Diss. ETH Nr. 10189

Spectral, ergodic and cohomological problems in dynamical systems

A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZÜRICH

for the degree of
Doctor of Mathematics

presented by
OLIVER KNILL
Dipl. Math. ETH
born Oktober 22, 1962
citizen of Appenzell/AI

accepted on the recommendation of
Prof. Dr. Oscar Lanford III, examiner
Prof. Dr. Alain-Sol Sznitman, co-examiner

1993

[Remark when putting online 2007: This is the introduction of the thesis together with the references only. The mathematica code, which had been written on a Next in Mathematica 2 or 3 is modified in the export part so that it runs today with Mathematica 5.2.]
Abstract e-collection of ETHZ. Entire thesis 410 pages Scanned PDF (OCR'd in 2012).

Abstract

$\bullet$ We showed for $\bf {K}=\bf {R},\bf {C}$ that every $SL(2,\bf {K})$ cocycle over an aperiodic dynamical system can be perturbed in $L^{\infty}(X,SL(2,\bf {K}))$ on a set of arbitrary small measure, so that the perturbed cocycle has positive Lyapunov exponents. We applied these results to show that coboundaries in $L^{\infty}(X,\bf {T}^1)$ or $L^{\infty}(X,SU(2))$ are dense.

$\bullet$ We proved, that Lyapunov exponents of $SL(2,\bf {R})$ cocycles over an aperiodic dynamical system depend in general in a discontinuous way on the cocycle. We related the problem of positive Lyapunov exponents to a cohomology problem for measurable sets.

$\bullet$ We showed the integrability of infinite dimensional Hamiltonian systems obtained by making isospectral deformations of random Jacobi operators over an abstract dynamical system. Each time- $ 1$ map of these so-called random Toda flows can be expressed by a $ QR$ decomposition.

$\bullet$ We proved that a random Jacobi operator $ L$ over an abstract dynamical system can be factorized as $ L=D^2+E$ , where $ E$ is real and below the spectrum of $ L$ and where $ D$ is again a random Jacobi operator but defined over a new dynamical system which is an integral extension. An isospectral random Toda deformation of $ L$ corresponds to an isospectral random Volterra deformation of $ D$ . The factorization leaded to super-symmetry and commuting Bäcklund transformations.

$\bullet$ We showed that transfer cocycles of random Jacobi operators move according to zero curvature equations, when the Jacobi operators are deformed in an isospectral way. We showed that every $SL(2,\bf {R})$ cocycle over an aperiodic system is cohomologuous to a transfer cocycle of a random Jacobi operator. We attached to any $ SU(N)$ random discrete gauge field a random Laplacian. From the density of states of this operator, it can be decided whether the gauge field has zero curvature or not. A cohomology result of Feldmann-Moore leaded to the existence of random Harper models with arbitrary space-dependent magnetic flux.

$\bullet$ We gave a new short integration of the periodic Toda flows using the representation of the Toda flow as a Volterra flow. We made the remark that the Toda lattice with two particles is equivalent to the mathematical pendulum. This gives a Lax representation for the mathematical pendulum. We rewrote the first Toda flow as a conservation law for the Green function of the deformed operator. We described a functional calculus for abelian integrals obtained by looking at an abelian integral on the hyperelliptic Toda curve as a Hamiltonian of a time-dependent Toda flow.

$\bullet$ By renormalisation of dynamical systems and Jacobi operators, we constructed almost periodic Jacobi operators in $\mbox{$\cal B$}$ $(l^2(\bf {Z}))$ having the spectrum on Julia sets $ J_E$ of the quadratic map $z \mapsto z^2+E$ for real $ E < -R$ with $ R$ large enough. The density of states of these operators is equal to the unique equilibrium measure $\mu_E$ on $ J_E$ . The set of so constructed random operators forms a Cantor set in the space of random Jacobi operators over the von Neumann-Kakutani system $T:X \rightarrow X$ , a group translation on the compact topological group of dyadic integers which is a fixed point of a renormalisation map in the space of dynamical systems. The Cantor set of operators is an attractor of the iterated function system built up by two renormalisation maps $\Phi^{\pm}_E$ .

$\bullet$ We proved that a sufficient conditions for the existence of a Toda orbit through a higher dimensional Laplacian $ L$ is that $ L$ is not a stationary point of the first Toda flow and that it is possible to factor $ L=D^2+E$ , where $ D$ is a random Laplacian over an integral extension. Random Laplacians appeared in a variational problem which has as critical points discrete random partial difference equations.

$\bullet$ We considered differential equations in $L^{\infty}(X)$ which form a thermodynamic limit of cyclic systems of ordinary differential equations. We considered also infinite dimensional dynamical systems describing the motion of infinite particles with pairwise interaction. The motion of random point vortex distributions can have a description as a motion of Jacobi operators.

$\bullet$ We constructed an analytic map $U: \bf {C}^4 \rightarrow \bf {C}^4$ , having a one-parameter family of two-dimensional real tori $S_{\gamma}$ invariant, on which $ U$ is the Standard map family $T_{\gamma}$ . We provided a rough qualitative picture of the dynamics of $ U$ and gave some arguments supporting the conjecture that the metric entropy of the Standard map $T_{\gamma}$ is bounded below by $\log(\gamma/2)$ .

$\bullet$ We introduced a generalized Percival variational problem of embedding an abstract dynamical systems in a monotone twist maps like for example the Standard map $S_{\gamma}$ . Using the anti-integrable limit of Aubry and Abramovici, we showed that there exists a constant $\gamma_0>0$ such that every ergodic abstract dynamical system $ (X,T,m)$ with metric entropy $h_m(T) \leq \log(2)$ and $\vert\gamma\vert \geq \gamma_0$ can be embedded in the twist map $S_{\gamma}$ . For such $\gamma$ , the topological entropy of $S_{\gamma}$ is at least $\log(2)$ . Using a generalized Morse index, the integrated density of states of the Hessian at a critical point, we proved the existence of uncountably many different embeddings of some aperiodic dynamical systems.

$\bullet$ We studied several cohomologies for dynamical systems: For a group dynamical system $ ($ $\mbox{$\cal R$}$ $ ,$ $\mbox{$\cal G$}$ $ )$ (the abelian group $\mbox{$\cal R$}$ is acting on the abelian group $\mbox{$\cal G$}$ by automorphisms) there is the Eilenberg-McLane cohomology. For a group dynamical system $(\bf {Z},$ $\mbox{$\cal G$}$ $ )$ we define a sequence of Halmos homology and cohomology groups. For an algebra dynamical system $(\bf {Z}^d,$ $\mbox{$\cal M$}$ $,{\rm tr})$ or for an group dynamical system $(\bf {Z}^d,$ $\mbox{$\cal G$}$ $ )$ , there is a discrete version of de Rham's cohomology.

$\bullet$ We studied the hyperbolic properties of bounded $SL(2,{\bf {R}})$ cocycles over a dynamical system. We investigated the relation between the rotation number of Ruelle for measurable matrix cocycles and the hyperbolic behavior of the cocycle. We showed that a cocycle is uniformly hyperbolic if and only if the rotation number is locally constant along a special deformation of the given cocycle. We proved that the spectrum of a cocycle acting on $L^2(X,\bf {C}^2)$ is the same as the Sacker-Sell spectrum.

Kurzfassung

$\bullet$ Wir zeigten, dass für $\bf {K}=\bf {R},\bf {C}$ jeder $SL(2,\bf {K})$ Kozyclus über einem aperiodischen dynamischen System im Raum $L^{\infty}(X,SL(2,\bf {K}))$ auf einer Menge von beliebig kleinem Mass gestört werden kann, so dass der gestörte Kocyclus einen positive Lyapunovexponenten hat. Wir wendeten dieses Resultat an, um zu zeigen, dass Koränder dicht in $L^{\infty}(X,\bf {T}^1)$ oder $L^{\infty}(X,SU(2))$ liegen.

$\bullet$ Wir bewiesen, dass Lyapunovexponenten von $SL(2,\bf {R})$ Kozyclen über einem aperiodischen dynamischen System im Allgemeinen unstetig vom Kozyclus abhängen. Wir finden eine Beziehung zwischen dem Problem, positive Lyapunovexponenten zu zeigen und dem Kohomolgieproblem für messbare Mengen.

$\bullet$ Wir zeigten die Integrabilität von unendlich dimensionalen Hamilton'schen Systemen, die durch isospektrale Deformation von zufälligen Jacobioperatoren über einem dynamischen System erhalten werden. Die Zeit-1-Abbildung von diesen sogenannten zufälligen Todaflüssen kann durch eine $ QR$ -Zerlegung ausgedrückt werden.

$\bullet$ Wir bewiesen, dass ein zufälliger Jacobioperator $ L$ faktorisiert werden kann als $ L=D^2+E$ , wo $ E$ reell und unterhalb des Spektrums von $ L$ liegt und $ D$ wieder ein zufälliger Jacobioperator über einer Integralerweiterung des alten Systems ist. Einer isospektralen zufälligen Todadeformation von $ L$ entspricht eine isospektrale zufällige Volterradeformation von $ D$ . Die Faktorisierung führte zu Supersymmetrie und kommutierenden Bäcklundtransformationen.

$\bullet$ Wir zeigten, dass Transferkozyklen unter isospektraler Deformation von zufälligen Jacobioperatoren sich gemäss Nullkrümmungsgleichungen bewegen. Wir zeigten, dass jeder $SL(2,\bf {R})$ -Kozyklus über einem aperiodischen dynamischen system kohomolog zu einem Transferkozyklus eines zufälligen Jacobiopertators ist. Wir adjungierten zu jedem diskreten $ SU(N)$ Eichfeld einen zufälligen Laplaceoperator dessen Zustandsdichte entscheidet ob das Eichfeld Krümmung Null hat oder nicht. Ein Kohomologieresultat von Feldmann und Moore führte zur Existenz von zufälligen Harpermodellen mit beliebigem ortsabhängigem magnetischem Fluss.

$\bullet$ Wir gaben eine neue kurze Integration des periodischen Toda Systems. Wir machten eine Bemerkung, die zu einer Lax-darstellung des mathematischen Pendels führte. Wir transformierten den ersten Todafluss als Erhaltungssatz für die Greenfunktion des deformierten Operators. Wir beschrieben ein Funktionalkalkül für abelsche Integrale indem ein abel'sches Integrale auf einer hyperelliptischen Todakurve als Hamiltonfunktion für ein zeitabhängiges Todasystem betrachtet wurde.

$\bullet$ Mittels Renormalisierung von dynamischen Systemen und Jacobioperatoren konstruierten wir fastperiodische Jacobioperatoren in $\mbox{$\cal B$}$ $(l^2(\bf {Z}))$ , die das Spektrum auf Juliamengen $ J_E$ der quadratischen Abbildung $z \mapsto z^2+E$ haben. Die Zustandsdichte von diesen Operatoren ist gleich dem eindeutigen Gleichgewichtsmass $\mu_E$ auf $ J_E$ . Die Menge der so konstruierten zufälligen Operatoren bilden eine Cantormenge im Raum der zufälligen Jacobioperatoren über dem von Neumann-Kakutani-system $T:X \rightarrow X$ , einer Gruppentranslation auf der kompakten topologischen Gruppe der dyadischen ganzen Zahlen die ein Fixpunkt einer Renormalisierungsabbildung im Raum der dynamischen Systeme ist. Die Cantormenge von Operatoren ist ein Attraktor eines iterierten Funktionensystems, das durch zwei Renormalisierungsabbildungen $\Phi^{\pm}_E$ gebildet wird.

$\bullet$ Wir zeigten, dass eine hinreichende Bedingung für die Existenz von einem Todafluss durch einen höher dimensionalen diskreten Laplaceoperator $ L$ ist, dass $ L$ nicht ein stationärer Punkt vom ersten Todafluss ist und dass es möglich ist, eine Faktorisierung $ L=D^2+E$ zu machen, wo $ D$ ein zufälliger Laplaceoperator über einer Integralerweiterung ist. Zufällige Laplaceoperatoren gibt es in einem Variationsproblem, dessen kritische Punkte durch partielle Differenzengleichungen beschrieben werden.

$\bullet$ Wir betrachteten Differentialgleichungen in $L^{\infty}(X)$ , die einen thermodynamischen Limes von zyklischen Systemen von gewöhnlichen Differentialgleichungen oder die Bewegung von unendlich vielen Teilchen mit Paarwechselwirkung beschreiben. Die Bewegung von zufälligen Punktwirbelkonfigurationen hat manchmal eine Beschreibung als Bewegung von einem Jacobioperator.

$\bullet$ Wir konstruierten eine analytische Abbildung $U: \bf {C}^4 \rightarrow \bf {C}^4$ , die eine einparametrige Familie von zwei-dimensionalen reellen Tori $S_{\gamma}$ invariant hat, auf denen $ U$ die Standardabbildung $T_{\gamma}$ ist. Wir machten eine grobe qualitative Beschreibung von $ U$ und gaben ein paar Argumente, die die Vermutung unterstützen, dass die metrische Entropie der Standardabbildung von unten durch $\log(\gamma/2)$ abschätzbar ist.

$\bullet$ Wir führten ein verallgemeinertes Percival'sches Variationsproblem zur Einbettung von abstrakten dynamischen Systemen in einer monotone Twistabbildung $S_{\gamma}$ ein. Unter Benützung des anti-integrablen Limes von Aubry und Abramovici zeigten wir, dass für grosse $\gamma$ jedes ergodische abstrakte dynamische System $ (X,T,m)$ mit metrischer Entropie $h_m(T) \leq \log(2)$ in die Standardabbildung $S_{\gamma}$ einbebettet werden kann. Für grosse $\gamma$ ist die topologische Entropie von $S_{\gamma}$ mindestens $\log(2)$ . Unter Benützung eines verallgemeinerten Morseindex bewiesen wir die Existenz von überabzählbar vielen verschiedenen Einbettungen von dynamischen Systemen.

$\bullet$ Wir studierten verschiedene Kohomologieen für dynamische Systeme: Für ein dynamisches System $ ($ $\mbox{$\cal R$}$ $ ,$ $\mbox{$\cal G$}$ $ )$ , wo eine abelsche Gruppe $\mbox{$\cal R$}$ auf einer abelschen Gruppe $\mbox{$\cal G$}$ operiert, gibt es die Eilenberg-McLane-Kohomologie. Im Falle $\mbox{$\cal R$}$ $=\bf {Z}$ definierten wir eine Folge von Halmos Kohomologie- und Halmos Homologie gruppen. Für ein dynamisches System, $(\bf {Z}^d,$ $\mbox{$\cal M$}$ $,{\rm tr})$ , wo $\bf {Z}^d$ auf der Algebra $\mbox{$\cal M$}$ mittels Spur erhaltenden Algebraautomorphismen operiert, definierten wir eine diskrete Version von de Rham's Kohomologie.

$\bullet$ Wir studierten das hyperbolische Verhalten von beschränkten, messbaren $SL(2,\bf {R})$ -Kozyklen über einem dynamischen System. Wir untersuchten die Relation zwischen der Rotationszahl von Ruelle für messbare Matrixkozyklen und dem hyperbolischen Verhalten der Kozyklen. Wir zeigten, dass ein Kozyklus uniform hyperbolisch ist, genau dann wenn die Rotationszahl konstant ist in einer Umgebung des Kozyklus. Wir betrachteten auch verschiedene Spektren von Matrixkozyklen und bewiesen, dass das Spektrum des Kozyklus als Operator auf $L^2(X,\bf {C}^2)$ gleich dem Sacker-Sell Spektrum ist.

Guide for the Reader

Each chapter is self-contained. This implies for example that some definitions occur several times and in different versions according to the generality needed at the corresponding places.
Because of the inhomogeneity of the themes, the notation is not uniform in different chapters throughout the theses.
Several chapters are already published and we plan to publish some parts separately hoping also to obtain in future more answers to some questions not yet solved. The already published chapters are partially updated to newer developments and sometimes, parts were added to the published version. Especially some illustrations and examples have been added.
Each chapter contains its own bibliography. There is also a collected bibliography at the end.
We added quite freely questions following each chapter. These questions help to keep organized the loose ends.
The current state of the chapters is the following:

$\bullet$ Chapter "Three problems":
This chapter introduces into three circles of problems which are treated in the theses.

$\bullet$ Chapter "Density results for positive Lyapunov exponents":
A large part of this chapter is published in [Kni 2]. We added a proof of a similar result for $SL(2,\bf {C})$ cocycles and updated some references.

$\bullet$ Chapter "Discontinuity and positivity of Lyapunov exponent":
This is essentially the version in [Kni 1]. We added two appendices concerning cohomology and a theorem of Baire.

$\bullet$ Chapter "Isospectral deformations of random Jacobi operators":
published in [Kni 3]. We added appendices in which the Thouless formula and integration of aperiodic Toda lattices is shown. These results in the appendices are known but not available in collected form. We added a short appendix about the definition of the continuous analogue of random Toda flows, the random KdV system.

$\bullet$ Chapter "Factorization of random Jacobi operators and Bäcklund transformations":
published in [Kni 4]. We clarified one aspect in the published proof of the fact that Bäcklund transformations are isospectral. We also added a short appendix about Bäcklund transformations for KdV flows.

$\bullet$ Chapter "Cohomology of $SL(2,\bf {R})$ cocycles and zero curvature equations for random Toda flows":
This chapter can be viewed as part III of the two previous chapters. It is quite inhomogeneous and leads to different topics like zero curvature equations, lattice gauge fields, cohomology of cocycles and Harpers Laplacian.

$\bullet$ Chapter "Some additional results for random Toda flows":
This is a continuation and unpublished work of research done for the previous chapters on random Jacobi operators. Also this chapter is not homogeneous and it has some loose ends.

$\bullet$ Chapter "Renormalisation of Jacobi matrices: Limit periodic operators having the spectrum on Julia sets":
A more compact version of this chapter will soon be submitted. This chapter is a continuation of the study of one dimensional Jacobi operators. Iteration of the factorization $ L=D^2+E$ constructed in [Kni 4] leads to "renormalisation" in the fiber bundle of Jacobi operators over the complete metric space of dynamical systems. Projecting this renormalisation to the spectrum of the operators leads to the quadratic map $z \mapsto z^2+E$ . Relations of random Jacobi operators with complex dynamical systems appear like for example that the density of states of the operators in the limit of renormalisation is the unique equilibrium measure on the Julia set or that the determinant $\det(L-E)$ is the Böttcher function for the quadratic map. We hope that results from the theory of iteration of rational maps will shed light on what happens at points where the renormalisation set up breaks down.

$\bullet$ Chapter "Isospectral deformations of discrete Laplacians":
We study higher dimensional Laplacians and isospectral deformations of such Laplacians. We show that the existence of a Dirac operator is related to isospectral deformations of Laplacians. We consider also random partial differential equations over a $\bf {Z}^d$ dynamical system. We use the anti-integrable limit of Aubry to prove the existence of such equations. An version of this chapter is planned to be presented in July 1993 at Leuven.

$\bullet$ Chapter "Infinite particle systems":
We study the idea to use ergodic theory in order to obtain the thermodynamic limit of infinite particle systems in one real or complex dimension. This generalization applies for ordinary differential equations describing particles. One idea is to get the particle coordinates $ q_n=q(T^nx)$ along the orbit of a point $ x$ . The thermodynamic limit is then a well defined ordinary differential equation in the Banach space of the coordinate field $ q$ . An other idea is to look at a vortex configuration in the complex plane as the spectrum of an operator and to describe the vortex flow as a flow of operators.

$\bullet$ Chapter "Embedding abstract dynamical systems in monotone twist maps":
this chapter [Kni 6] was submitted in October 1992 to Inventiones and in December 1992 to Ergodic Theory and Dyn. Syst. In both cases the paper was not accepted. We take the opportunity to comment on the results. The main point of the chapter is that it allows to give an quantitative explicit bound on the topological entropy for monotone twist maps. This is a new application of the anti-integrable limit of Aubry and is (according to our opinion) an interesting result. It has not been obtained by other methods and other approaches (like finding homoclinic points) for positive topological entropy are more complicated. The use of the generalized Morse index in the paper is also new. The multiplicity result of uncountably many different embeddings has not been obtained by other methods.

$\bullet$ Chapter "An analytic map containing the standard map family":
This chapter was submitted in October 1992 to Nonlinearity [Kni 5]. We formulate a quantitative conjecture about the metric entropy of the standard map and show that the whole standard map family can be embedded in one complex analytic map of $\bf {C}^4$ . We begin a qualitative study of the map.

$\bullet$ Chapter "Cohomology of dynamical systems":
In this chapter we discuss some cohomological constructions for abstract dynamical systems with the aim to build algebraic invariants of the systems. We think that interesting research in this direction is still possible and necessary. The chapter illustrates that the definition of the cohomologies is quite easy but that the explicit computation of the cohomology groups seems to be very difficult even in the simplest cases.

$\bullet$ Chapter "Nonuniform and uniform hyperbolic cocycles":
In this chapter, we discuss the relation of uniform and nonuniform hyperbolicity for cocycles. We study also the relation between Lyapunov exponents and rotation numbers and the relation between Lyapunov exponents and spectra of cocycles treated as operators.
This chapter was written in an early stage of the theses and does not contain so much new material. It can be viewed as an appendix to the first two chapters on matrix cocycles and one can find for example detailed proofs of a theorem of Ruelle and a theorem of Wojtkowsky in the special case of measurable $SL(2,\bf {R})$ cocycles.

The basic objects

The basic (sometimes hidden) mathematical structure appearing in all the chapters is a non-commutative von Neumann algebra $\mbox{$\cal X$}$ obtained as a crossed product of the von Neumann algebra $\mbox{$\cal M$}$ $=L^{\infty}(X,M(N,\bf {C}))$ with the $\bf {Z}^d$ - dynamical system

$\displaystyle (X,T_1,T_2, \dots, T_d,m) \; ,$

where $T_1, \dots, T_d$ are commuting automorphisms of the Lebesgue probability space $ (X,m)$

$\fbox{ \parbox{12cm}{ The basic question is to find {\bf spectral}, {\bf ergodic... ...\mbox{$\cal X$}$\ and relate them to invariants of the dynamical system. \\ }}$

$\bullet$ Spectra are obtained by choosing a representation of $\mbox{$\cal M$}$ in an algebra of operators and taking as the spectrum of an element $ A$ in $\mbox{$\cal M$}$ either the set of complex values $ z$ such that $ z-A$ is not invertible or to define a spectrum by taking the set of points $ z$ , where $z \cdot A$ is not invertible. Choosing special cocycles (like for example the Jacobean of a map,or the Koopman operator associated to an abstract dynamical system) leads to invariants of the dynamical system.

$\bullet$ Ergodic invariants are numbers obtained by ergodic averaging. Examples are the Lyapunov exponent, the rotation number or the total curvature of a cocycle or field. Choosing special cocycles leads to invariants of the dynamical system like for example the entropy or the index of an embedded system.

$\bullet$ (Co)homology groups or Moduli spaces of conjugacy classes of a subset in $\mbox{$\cal X$}$ give algebraic informations about a dynamical system. Examples are cohomology groups $\mbox{$\cal H$}$ $ ^1(T,G)$ defined by the quotient of all cocycles with values in the group $ G$ modulo the space of coboundaries.

The elements in $\mbox{$\cal X$}$ we are going to study are:

$\bullet$ $d=1,N=2, A \tau$ is called a matrix cocycle over the abstract dynamical system $ (X,T,m)$ . Such cocycles appear as transfer cocycles of one dimensional discrete Schrödinger operators, as Jacobeans of diffeomorphisms on compact two dimensional manifolds and which are also called weighted composition operators or weighted translation operators.

$\bullet$ $\sum_{i=1}^d A_i \tau_i$ are called one-forms, or connections or non-abelian gauge fields.

$\bullet$ $d=1,N=1, L=a\tau+(a\tau)^*+b$ is a discrete version of a one dimensional Schrödinger operator. Such operators serve as models in solid state physics. They appear also as Hessians of variational problems in monotone twist maps. For general $ N$ , they are called Jacobi operators on a strip.

$\bullet$ $L=\sum a_i \tau_i + (a_i \tau_i)^* + b$ is a discrete version of a Laplace operator. We call it random Laplace operator. We allow also $ N >1$ .

The role of Lyapunov exponents

The title of the theses could also be "Some topics about Lyapunov exponents" because Lyapunov exponents will appear in almost all chapters, sometimes with other names. They are connected with

$\bullet$ the "entropy" of a dynamical system,
$\bullet$ the "determinant" of a random Jacobi operators,
$\bullet$ "integrals" of random Toda flows,
$\bullet$ the "Hamiltonian" used for the interpolation of Bäcklund transformations,
$\bullet$ the "energy" of a random Coulomb gas in two dimensions,
$\bullet$ the potential theoretical "Green function" of some Julia sets,
$\bullet$ the logarithm of a "spectral radius" of a weighted composition operator,
$\bullet$ some "gauge invariants" of random gauge fields.

The intuitive idea

We will formulate in the first chapter three problems which should give a first comment on the key words "ergodic invariants, spectral invariants, cohomological invariants" in the title.

A guiding idea for the whole theme is the following analogy between differential topology on manifolds and the ergodic theoretical concepts treated here.

If we think of the probability space $ (X,m)$ as a manifold, the group action defines a geometry on this space. The orbit of a point $ x$ is a lattice is playing the role of a chart at the point and the set of all orbits serves as an atlas.

$L^{\infty}(X,M(N,\bf {C}))$ or a multiplicative subgroup $L^{\infty}(X,G)$ corresponds to a fiber bundle. The crossed-product $\mbox{$\cal X$}$ is an operator algebra playing the role of differential operators and contains objects like Laplacians or connections.

The classification of differentiable manifolds is analogous to the classification of group actions. Spectral problems of differential operators, numerical or cohomological invariants of manifolds are analogous to spectral problems of random operators, ergodic and cohomological invariants of dynamical systems.

Three Problems

We give three examples of problems which should illustrate some topics of this theses.

The first problem is the problem of proving positive metric entropy in a given conservative dynamical system.
The second problem is the problem of determining the spectrum of random Schrödinger operators.
The third problem is to determine a cohomology group for abstract dynamical systems.

All three problems are related to the problem of calculating Lyapunov exponents of matrix cocycles over an abstract dynamical system.

An ergodic problem: positive metric entropy

"Is there positive metric entropy ("chaos") in a given conservative dynamical system?"

One of many definitions of "chaos" in a Hamiltonian system is the property of having positive metric entropy. Hamiltonian systems with this property show sensitive dependence on initial conditions on a set of positive measure. Moreover, one can find quantities accessible to measurements which are actually independent random variables. The system could be used as a true random number generator !

(We refer to the Les Houches Lecture of Lanford [Lan 83], the Bernard Lecture [Rue 90] or the Lezioni Lincee [Rue 87] of Ruelle for an introduction into some of the topics.)

From the physical point of view, there is evidence that chaos is the rule and zero metric entropy is the exception. The entropy has been measured in many systems and found to be positive. It seems, however, that mathematical proofs for chaos are difficult. It could even be that most measurements show numerical artifacts and that positive metric entropy is the exception.

There are milder requirements for a system to belong to the fashion class of "chaotic systems". One of these is positive topological entropy which is easier to prove. For topological dynamical systems, there is a definition of chaos due to Devaney [Dev 89] which requires that the homeomorphism of the metric space is (i) transitive and (ii) that periodic orbits are dense. (A third requirement of Devaney, sensitive dependence on initial conditions, turned out to follow from the requirements (i),(ii) if the system is not itself periodic [Ban 92].)

We want to insist on positive metric entropy as the more relevant quantity for Hamiltonian systems. It is based on the belief, that physically significant quantities for Hamiltonian systems must be accessible on sets of positive measure. Also from a mathematical point of view it is not satisfactory to use a quantity like topological entropy from the category of topological dynamical systems when having a natural invariant measure, the Lebesgue measure. (Of course, invariant sets like periodic orbits or horse-shoes, which have in general zero measure, can also have consequences for sets of positive measure. For example, invariant curves of zero measure can form barriers in a two dimensional phase space and prevent ergodicity.)

The answers to the following general problems in classical mechanics are not known. (We assume implicitly that the Hamiltonian systems considered have a compact energy surface and that the entropy is understood with respect to the flow on such a surface.)

$\bullet$ How does one decide if a given finite dimensional Hamiltonian system has positive metric entropy or not?

$\bullet$ How big is the class of Hamiltonian systems having positive metric entropy? Does positive metric entropy occur generically?

$\bullet$ Does every non-integrable Hamiltonian system have positive metric entropy?

$\bullet$ Does there exist Hamiltonian systems which have positive topological entropy but zero metric entropy?

$\bullet$ Does there exist a Hamiltonian system in which true coexistence occurs? ([Str 89]) True coexistence means that a part of phase space is the union of two flow-invariant dense subsets both having positive Lebesgue measure such that the metric entropy of the flow is zero on the first subset and positive on the other subset.

The main obstacle in solving the above problems is that proving positive metric entropy is a hard problem. Because a formula of Pesin shows that in many cases the entropy is the sum of the integrated positive Lyapunov exponents, the problem of positive metric entropy is often equivalent to prove that there are positive Lyapunov exponents on a set of positive measure.

There are several examples, where positive metric entropy is conjectured but not proven (compare [Str 89]). From special interest is the example of our solar system, where positive metric entropy was also measured [Las 90]. (See also the review article [Wis 87] for chaotic motion in the solar system.) We will give now a list of examples which include discrete Hamiltonian systems (which are mappings) and classical Hamiltonian systems (which are flows). We consider also conservative dynamical systems, flows or maps which leave the Lebesgue measure invariant but which need not to be Hamiltonian.

For the Chiricov mapping Chiricov map or Standard mapping on the torus $\bf {T}=\bf {R}^2/(2 \pi \bf {Z})$

$\displaystyle T_{\gamma}:\left[ \begin{array}{c} x \\ y \\ \end{array} \right] ... ...egin{array}{c} x+y+\gamma \sin(x) \\ y+\gamma \sin(x) \\ \end{array} \right]$

the entropy is measured to be greater or equal then $\log(\vert\gamma/2\vert)$ . Some orbits of the map are shown with the following Mathematica program which produced a little "film" which shows the situation for larger and larger "coupling constant" $\gamma$ .

T[{x_Real,y_Real},g_Real]:=N[Mod[{x+y+g*Sin[x],y+g*Sin[x]},2 Pi]];
Orbit[p0_,n_Integer,g_Real]:=Module[{},S[p_]:=T[p,g];NestList[S,p0,n-1]];
OrbitSet[k_Integer,n_Integer,g_]:=Module[{s={}},
  Do[s=Union[s,Orbit[{0.0,N[2Pi*((Sqrt[5]-1)/2+(i-1)/k)]},n,g]],{i,k}];s];
Pict[k_,m_,g_]:=ListPlot[OrbitSet[k,m,g],PlotRange->{{0,N[2Pi]},{0,N[2Pi]}},
  DisplayFunction->Identity,Frame->True,Axes->False,FrameTicks->None,
  FrameLabel->{g,"","",""}];
Film[kk_,mm_,ll_]:=Table[Pict[kk,mm,0.0+N[(jj-1)/4]],{jj,ll}];
Show[GraphicsArray[Partition[Film[7,300,12],3]],
  DisplayFunction->$DisplayFunction,Frame->False];

Scan of the actual thesis introduction

If one considers the Jacobian $dT_{\gamma}$ of the map $T_{\gamma}$ as a cocycle over the dynamical system $({\bf {T}},T_0,m)$ and calculates the Lyapunov exponent of this cocycle, one gets indeed with a method developed by M.Herman) the lower bound $\log(\vert\gamma/2\vert)$ . But it is an unsolved problem whether there exists a value $\gamma$ such that the metric entropy is positive for the standard map $S_{\gamma}$ . (This entropy problem has been mentioned already in [Spe 86]). For the topological entropy more is known: there is a result of Angenent [Ang 92] which states that a $C^{1+\epsilon}$ twist diffeomorphism of the annulus has either positive topological entropy (and therefore a transversal homoclinic point) or else has invariant circles for any rotation number in the twist interval. In [Ang 90] is shown that for $\gamma>1$ , there must be positive topological entropy. Fontich has also shown [Fon 90], that for $\gamma>0$ the standard map has a heteroclinic point and therefore a horse-shoe embedded and one has therefore positive topological entropy. In the chapter "Embedding of abstract dynamical systems in monotone twist maps", we will show that for $\gamma$ large enough the topological entropy of the Standard map is at least $\log(2)$ .

Among monotone twist mappings, billiards have been investigated extensively. There is no known example of a strictly convex smooth billiards with positive metric entropy. (On the other hand one has also the unsolved question, which tables produce integrable billiard maps. Guillmin conjectures that every billiards which is not integrable must be an ellipse ([Gui 87]). An other open question is whether every billiards with zero metric entropy is an ellipse.) There are known classes of examples of not smooth billiards having positive metric entropy. The most famous example is Bunimovitch's stadium billiards (see [Woj 86] or [Don 91] and references therein for newer results). A candidate for a smooth billiards with positive metric entropy is the Robnik billiards. It is defined by the curve

$\displaystyle \gamma: t \ \mapsto \left[ \begin{array}{c} \cos(t)+\gamma \cos(2 t) \\ \sin(t)+\gamma \sin(2 t) \\ \end{array} \right] \; ,$

where $\gamma \in (0,1/4)$ . One measures positive metric entropy with values like for example $\gamma=0.2$ . Other candidates are billiards, where the curve consists of arcs (see [Hay 87]) or the Benettin-Strelcyn Billiard [Ben 78],[Hen 83].

A far as we know, no smooth dual billiards map with positive metric entropy has been constructed. The dual billiards or exterior billiards is a mapping defined outside a convex curve in the plane. It preserves Lebesgue measure. Application of KAM methods [Dua 82] (III-12) show that for smooth dual billiards curves, there exist invariant curves of the dynamics. Therefore, the dynamics is defined on regions of finite Lebesgue measure and the question of positive metric entropy is well defined. Tabatchnikov [Tab 93] has recently shown that if two dual billiards maps are commuting then the tables are conformal ellipses. Dual billiards at polygons is already very interesting and there are many open questions (see [Viv 87], [Gut 92] in this case and general references).

The following Mathematica program allows the numerical calculation of a dual billiards with arbitrary real analytic table. The program calculates and plots a picture of 20 orbits each consisting of 1000 points.

a={1.0,0.0,0.0,0.0,0.0,0.0,0.01,0.001};
r[t_]:=Sum[a[[n]] Cos[(n-1)*t],{n,Length[a]}];
rdot[t_]:=Sum[-a[[n]]*(n-1)*Sin[(n-1)*t],{n,2,Length[a]}];
BilliardMap[{x_,y_}]:=Module[{psi=Arg[x+I*y],eta},
eta=t/.FindRoot[ (r[t]*Cos[t]-x)*(rdot[t]*Sin[t]+r[t]*Cos[t])==
         -(r[t]*Sin[t]-y)*(-rdot[t]*Cos[t]+r[t]*Sin[t]),{t,psi+Pi/2}];
  {-x+2*r[eta]*Cos[eta],-y+2*r[eta]*Sin[eta]}];
T=ParametricPlot[{r[t]*Cos[t],r[t]*Sin[t]},{t,0,2 Pi}];
Orbit[p_,n_]:=NestList[BilliardMap,p,n];
OrbitSet[n_,m_]:=Module[{s={}},
  Do[s=Join[s,Orbit[{1.0+i*0.05,0.0},n]],{i,m}];s];
OrbitSetPict[n_,m_]:=ListPlot[OrbitSet[n,m],DisplayFunction->Identity];
JoinedPict[n_,m_]:=Show[{T,OrbitSetPict[n,m]},
   PlotRange->All,Frame->True,Axes->False];
Show[JoinedPict[1000,20],Frame->True,AspectRatio->1];

and here is the picture

There are also interesting monotone twist maps in the plane $\bf {R}^2$ of the form

$\displaystyle T=T_f: \bf {R}^2 \rightarrow \bf {R}^2$

$\displaystyle \left[ \begin{array}{c} x \\ y \end{array} \right] \mapsto \left[ \begin{array}{c} f(x)-y \\ x \end{array} \right] \; ,$

where is a continuous function $f:\bf {R}\mapsto \bf {R}$ . Examples are the Henon map , the Lozi map $f(x)=1+a\vert x\vert$ or the map given by $f(x)=\sqrt{a^2+x^2}$ . The entropy question makes sense when these maps are restricted to a measurable invariant subset of $\bf {R}^2$ which has finite nonzero Lebesgue measure. For $f(x)=\sqrt{a^2+x^2}$ , Aharonov and Elias [Aha 90] have shown the existence of invariant curves far away (even for a more general case when is the composition of two such maps with possibly different ). It seems however (a conjecture of H.Cohen communicated by Y.Colin-de Verdier to J.Moser April 10 1993) that the map with $f(x)=\sqrt{a^2+x^2}$ is integrable! (For there is an additional integral $F(x,y) = y+\vert y-\vert x\vert\vert +\vert x-\vert y-\vert x\vert\vert\vert +... ...rt\vert\vert +\vert x-\vert y\vert+\vert y-\vert x-\vert y\vert\vert\vert\vert$ ). For the Lozi map, it is known that it has positive metric entropy for example for [Dev 84]. Nothing about the metric entropy seems to be known for the conservative Henon map.
A promising strategy to find true coexistence is to start from a system with positive metric entropy and to move into a region with zero entropy. One expects then to pass a region of true coexistence. Coexistence of stable (existence of an elliptic fixed point) and unstable (positive metric entropy) behavior has been constructed by Przytycki [Prz 82] by constructing a real analytic arc starting at an Anosov system. It is however not known if this example satisfies true coexistence. An other candidate of a perturbation of an Anosov map is

$\displaystyle T_f:\left[ \begin{array}{c} x \\ y \\ \end{array} \right] \mapsto... ...[ \begin{array}{c} 2x+y \\ x+y+ \gamma \sin(2x+y) \\ \end{array} \right] \; .$

It is not known if one can deform this Anosov map to a system which has zero metric entropy. It would also be interesting to know what happens at the boundary, where the system looses uniform hyperbolicity.
Among Hamiltonian systems, there are many candidates with two degrees of freedom which are believed to have positive metric entropy:

$\bullet$ The Henon Heils system : [Hen 64]

$\displaystyle H(p,q)=\frac{1}{2}(p_1^2+p_2^2+q_1^2+q_2^2)+q_1^2q_2-\frac{1}{3}q_2^2 \;$

on the energy surface with is a famous example of a Hamiltonian system with two degrees of freedom.

$\bullet$ The Störmer problem (see [Bra 81]) :

$\displaystyle H(p,q)=\frac{1}{2}(p_1^2+p_2^2)+\frac{1}{2}(\frac{1}{q_1}- \frac{q_1}{(q_1^2+q_2^2)^{3/2}})^2$

for energies is an example of great physical interest because it describes the motion of electrons and protons in the van Allen radiation belts. It would be desirable to know more about this system. There are mathematical proofs that particles can be trapped for ever [Bra 81], but one still does not know why the actual existing van Allen belts around the earth are so stable. Numerical experiments indicate that the system shows chaotic behavior. Braun has proved that a related discrete model problem, a so called linked twist map has homoclinic points [Bra 81a]).

$\bullet$ The Contopoulos,Barbanis system [Con 89]

$\displaystyle H(p,q)=\frac{1}{2}(p_1^2+p_2^2+q_1^2+q_2^2)+q_1^2q_2-q_1 q_2^2 \; .$

or the Caranicolas-Vozikas system [Car 87]

$\displaystyle H(p,q)=\frac{1}{2}(p_1^2+p_2^2)+ q_1^4+q_2^4+2 \gamma q_1^2 q_2^2$

for , $\gamma=6$ , or the Sahos-Bountis system

$\displaystyle H(p,q)=\frac{1}{2}(p_1^2+p_2^2)+ \frac{1}{2}(q_1^2q_2^2+ \gamma (q_1^2+q_2^2)) .$

are interesting because of the simplicity of their Hamiltonians.

$\bullet$ Unequal mass Toda system:

$\displaystyle H(p,q)=\frac{1}{2}(\frac{p_1^2}{m_1}+ \frac{p_2^2}{m_2})+e^{q_1-q_2}+e^{q_2-q_1}$

for $m_1 \neq m_2$ . For the system is integrable and is related to the physical pendulum.

$\bullet$ The planar isosceles 3 body problem (see for example [Dev 80])

$\displaystyle H(p,q)=\frac{p_1^2}{m_1} + \frac{(2 m_1+m_3) p_2^2}{4 m_1 m_3} -\frac{m_1^2}{q_1}-\frac{4 m_1 m_3}{(q_1^2+4 q_2^2)^{1/2}} \; .$

Here are the masses of the body. is the distance from to which are lying symmetric with respect to the axes. is the distance from the center of mass of and to .
Interesting Hamiltonian systems arise from geodesic flows. The question is then which smooth Riemannian metrics on a compact manifold have positive metric entropy for the geodesic flow. Donney [Don 88],[Don 88] proved that on every compact orientable surface, there exists a $C^{\infty}$ Riemannian metric, for which the geodesic flow is ergodic and has positive metric entropy.
There are interesting cyclic systems of differential equations which are candidates for systems having positive metric entropy.

$\bullet$ The Orszag-McLaughlin flow is given by the differential equation

$\displaystyle \dot{x}_i=a \cdot x_{i+1} x_{i+2}+b \cdot x_{i-1} x_{i-2} + c \cdot x_{i+1}x_{i-1} \; ,$

where the index is taken modulo with $n \geq 3$ . If one takes and $abc \neq 0$ , this system preserves the Lebesgue measure and leaves invariant the spheres

$\displaystyle S_r= \{ \sum_{i=1}^n x_i^2=r \} \; .$

The measurements indicate that the dynamics restricted to such spheres is chaotic.

$\bullet$ The Arnold-Beltrami-Childress flow or ABC-flow (see [Zha 93] for more information) is a flow on $\bf {T}^3$ defined by

$\displaystyle \dot{x}$ $\displaystyle =$ $\displaystyle A \sin(z) + C \cos(y)$

$\displaystyle \dot{y}$ $\displaystyle =$ $\displaystyle B \sin(x) + A \cos(z)$

$\displaystyle \dot{z}$ $\displaystyle =$ $\displaystyle C = \sin(y) + B \cos(x) \; ,$

where are nonzero constants. More generally with given real numbers $a_i \in \bf {R}$ , a system of differential equations is given by

$\displaystyle \dot{x}_i=a_{i-1} \sin(x_{i-1}) +a_{i+1} \cos(x_{i+1}) \; ,$

where $x_i \in \bf {R}/\bf {Z}$ and is taken modulo . This system preserves Lebesgue measure on the torus ${\bf {T}}^n$ .
The heavy top and its higher dimensional generalizations. Given $M \in sl(n,{\bf {R}})$ , $B \in so(n,{\bf {R}})$ and $J=J^* \in GL(n,{\bf {R}})$ . For the differential equation

$\displaystyle \dot{L}=[B,L]+M \;$

generalizes the motion of the heavy top in any dimension. For one obtains the motion of the free dimensional top which is an integrable system and can be viewed as a geodesic flow on the Lie group $SO(n,\bf {R})$ (see [Arn 89]). For $M \neq 0$ , one expects to have positive metric entropy in general.
The Hamiltonian for vortices in the complex plane $\bf {C}$ is

$\displaystyle H(z)=\frac{1}{2\pi i} \sum_{i<j} \log\vert z_i-z_j\vert \; .$

For $n \leq 3$ the system is integrable. For $n \geq 4$ one expects that the dynamics

$\displaystyle \dot{q}=\frac{\partial}{\partial p} H, \; \dot{p}=-\frac{\partial}{\partial q} H$

has positive metric entropy. Point vortices can also be defined on any two dimensional manifold. Interesting is the situation on the cylinder which is also called the periodic Karman vortex street. The Hamiltonian is in this case

$\displaystyle H(z)= \sum_{i<j} \log (\sin \vert z_i-z_j\vert) \; .$

We summarize: for conservative dynamical systems the problem of positive metric entropy is often reduced to the decision whether the highest Lyapunov exponents for symplectic cocycles is positive or not (the flow case can be reduced by a time-one map or a Poincaré map to the case when time is discrete). Calculating the highest Lyapunov exponent of a general measurable symplectic cocycle over an abstract dynamical system is an example of an ergodic problem.

A spectral problem: random Jacobi operators

Does a given 1-dimensional Schrödinger operator of an electron have absolutely continuous spectrum ("good conductivity")?

In classical quantum mechanics, the evolution of a system is governed by a Hamiltonian $ L$ which is a self adjoint operator on a Hilbert space. The evolution of a wave function $ u$ is given by the Schrödinger equation $i h \dot{u}=Lu$ . An important problem is to calculate and analyze the spectrum of the Hamiltonian because many physical properties of the system are determined by the spectrum. Because the spectrum is accessible by measurements, one would also like to solve the inverse problem, namely to find out the Hamiltonian of the system from the spectrum.

These mathematical problems are not easy even for the motion of a particle in one dimension in a given external field. In such a one body approximation one neglects the interaction of the electrons. One often considers the so called tight binding approximation, where the continuum is replaced by a lattice. This is technically more simple and the model is commonly used for describing the electronic properties of disordered media.

An important qualitative spectral problem is the question whether the spectrum is absolutely continuous or not. For one-dimensional Laplacians or Laplacians on the strip, this question can be reduced to a question about positive Lyapunov exponents for symplectic cocycles.

A main problem is: Given a random Schrödinger or Jacobi operator $ L$ .

What is the spectrum of ?
What type of spectrum (point, singular, or absolutely continuous) does occur?
What spectra do occur generically? Is there generically no absolutely continuous spectrum?

There is an inverse spectral problem which is however closely related to the above spectral problem. It can be formulated as:

$\bullet$ Can one describe the set of all Jacobi operators which are unitarily equivalent or the set of operators which have the same density of states. Can one reconstruct the isospectral set back from the spectrum. How large is the set of operators which one can reach by isospectral deformations?

Similarly to the positive entropy question, which has the same mathematical problem in the background, there are several examples, where absence of absolutely continuous spectrum is conjectured but not proved. The general belief is that for high disorder, there is no absolutely continuous spectrum any more. A more precise formulation of this conjecture could be:

Given any aperiodic dynamical system $ (X,T,m)$ such that $ T^n$ is ergodic for each $n \in \bf {Z}$ . Given a one- parameter family of random Schrödinger operators

$\displaystyle (L_gu)_n=u_{n+1}+u_{n-1}+g \cdot V_n u_n$

with non-constant potential $V \in L^{\infty}(X,\bf {R})$ and with $ V_n=V(T^nx)$

. Does there exist $g \in {\bf {R}}$ , such that $ L_g$

has no absolutely continuous spectrum?

We list now some classes of operators.

$\bullet$ The Anderson model

$\displaystyle (Lu)_n=u_{n+1}+u_{n-1}+V_n u_n \; ,$

where

are non-constant independent identically distributed random variables. Fürstenberg's theorem assures positive Lyapunov exponents for all $E \in \bf {R}$ and there is no absolutely continuous spectrum.

$\bullet$ Almost periodic operators.
Let $ T$ be an ergodic translation of a compact topological group $ X$ and $ a,b$ two continuous real-valued functions $X \mapsto \bf {R}$ . Given $x \in X$ , define $ a_n=a(T^nx),b_n=b(T^nx)$ . The operator

$\displaystyle (Lu)_n=a_n u_{n+1}+a_{n-1} u_{n-1} +b_n u_n \;$

is called almost periodic.

Example. If the group $ X$ is the circle and $ T$ is an irrational rotation $x \mapsto x+\alpha$ and $V(x)= \cos(x)$ , one obtains the almost Mathieu operator

$\displaystyle (L(x)u)_n=u_{n+1}+u_{n-1}+ \gamma \cos(x) \; .$

This is a famous model for an electron in a quasi-crystal. About the spectrum $\sigma=\sigma_{pp} \cup \sigma_{sc} \cup \sigma_{ac}$ is known that for $\lambda<4/(2+\pi/\sqrt{5})$ the absolutely continuous spectrum $\sigma_{ac}$ is not empty. For almost all $\alpha$ , the measure of the absolutely continuous spectrum is $4-\gamma$ . This was proved recently by Last [Las 93] following previous work of Avron, van Mouche and Simon [Avr 90]. The general conjecture of Aubry and André that this should hold for any irrational $\alpha$ is still not proved. For large $\gamma$ , one knows that in the case of Diophantine $\alpha$ , there is point spectrum [Fro 90] and that for Liouville numbers $\alpha$ and any $\gamma>2$ , there is purely singular continuous spectrum (see [Cyc 87]). For any irrational $\alpha$ , there is no absolutely continuous spectrum for $\gamma>2$ . In general, the question is open, whether for irrational $\alpha$ and all $\gamma$ , the spectrum is a Cantor set (Martini problem). In the critical Hofstadter case $\lambda=2$ , which is also called Harper's model, Last has recently shown [Las 93] that the spectrum is in this case a zero measure Cantor set for almost all $\alpha$ . Plotting the family of spectra parameterized by $\alpha$ gives the Hofstadter butterfly. A review with other developments can be found in [Bel 92].

A numerical illustration. The periodic functions $ a,b$ on $\bf {T}^1$ are fixed and the spectra of the operators $L=a\tau_{\alpha}+(a\tau_{\alpha})^*+b$ over dynamical system $(\bf {T}^1,x \mapsto x+\alpha,dx)$ are calculated with the following program:

a[alpha_,n_]:=Table[N[2+0.01*Sin[k*alpha 2 Pi]],{k,n}];
b[alpha_,n_]:=Table[N[  2*Cos[k*alpha 2 Pi]],{k,n}];
m[i_,n_]:=Mod[i-1,n]+1;
d[k_,l_,n_]:=IdentityMatrix[n][[m[k,n],m[l,n]]];
JacobiMatrix[a_,b_]:=Module[{n=Length[a]},
 Table[d[k,i,n]*b[[k]]+d[k,i+1,n]*a[[i]]
      +d[k,i-1,n]*a[[m[i-1,n]]],{k,n},{i,n}]];
Spec[a_,b_]:=Sort[Eigenvalues[JacobiMatrix[a,b]]];
anti[a_]:=Block[{n=Length[a]},Table[a[[k]]-2*d[k,n,n]*a[[k]],{k,n}]];
AntiSpec[a_,b_]:=Sort[Eigenvalues[JacobiMatrix[anti[a],b]]];
DoubSpec[a_,b_]:=Sort[Join[Spec[a,b],AntiSpec[a,b]]];
SpecInterv[a_,b_]:=Partition[DoubSpec[a,b],2];
Hofstadter[m_,n_]:=Block[{p={},S,alpha=0.0},
Do[S=SpecInterv[a[alpha,n],b[alpha,n]];
   Do[p=Append[p,Line[{{S[[i]][[1]],alpha},{S[[i]][[2]],alpha}}]],
   {i,Length[S]}];alpha=alpha+1/m,{m+1}]; Show[Graphics[p]] ];
Hofstadter[200,23];

which produces the following picture which shows the spectrum of such an operator changes when the parameter $\alpha$ (which is increasing along the x-axis) of the dynamical system is varied.

[Note added, when putting this introduction online in 2007: a better picture of the butterfly in color can be found here.]

$\bullet$ Operators generated by substitutions.
A substitution dynamical system (we follow [Hof 93]) is defined by a map $ S$ from a finite alphabet $ A$ into the set of finite words $ A^*$ built by $ A$ . This substitution generates a fixed point $\tilde{x} \in A^{\bf {N}}$ of the map $ S$ if there exists a symbol $a \in A$ such that $ S a$ begins with $ a$ . Take any word $\overline{x} \in A^{\bf {Z}}$ which coincides with $\tilde{x}$ on the positive integers and form the set $ X$ of all limit points (in the product topology) of $T^n\overline{x}$ , where $ T$ is the shift. This gives a dynamical system $ (X,T)$ which is uniquely ergodic and minimal. The potential $ V$ for the Jacobi operator is $ V(x)=x_0$ . The spectrum is expected to be singular continuous in general and having zero Lebesgue measure. This has been proved for the Thue-Morse systems defined by $ S(a)=ab$ , $ S(b)=ba$ or Fibonacci sequences defined by , $ S(b)=a$ and other examples. Kotani [Kot 89] dealed in a more general context with potentials over an ergodic dynamical system which take values only in a finite target space. He proved that in such a case, there is no absolutely continuous spectrum. See [Hof 93] or [Ghe 92] for references and recent results.

$\bullet$ Operators which are second variations of twist mappings
If $ l(x,x')$ is a generating function of a twist map and $T:X \mapsto X$ is a dynamical system embedded in the twist map by a measurable function

$\displaystyle q: X \mapsto \bf {T}$

which satisfies

$\displaystyle l_{1}(q(x),q(Tx))+l_{2}(q(T^{-1}x),q(x))=0 \; .$

The operator

$\displaystyle (L(x)u)_n=a_n u_{n+1}+a_{n-1} u_{n-1} +b_n u_n \; ,$

with

$\displaystyle a(x)$	$\displaystyle =$	$\displaystyle l_{12}(q(x),q(Tx)) \; ,$
$\displaystyle b(x)$	$\displaystyle =$	$\displaystyle l_{11}(q(x),q(Tx))+l_{22}(q(T^{-1}x),q(x))$

is the second variation of a variational problem. It would be interesting to know what are the spectra of these Hessians.

We summarize: for one-dimensional discrete Schrödinger operators (on the strip), the problem of existence of absolutely continuous spectrum is reduced to the decision whether the highest Lyapunov exponent for a one parameter family of symplectic cocycles is positive or not. Calculating the spectrum of random operators or cocycles over an abstract dynamical system is an example of a spectral problem.

A cohomological problem: cohomology of measurable sets

" What is the cohomology group defined as the group of measurable sets of a probability space modulo the sets of the form $Z(T) \Delta Z$ , where is an automorphism of the probability space? "

Given an aperiodic ergodic automorphism $ T$ of the Lebesgue space $ (X,$ $\mbox{$\cal A$}$ $ ,m)$ . The measurable sets $Y \in$ $\mbox{$\cal A$}$ are called cocycles and form an abelian group with multiplication $\Delta$ . This group has the subgroup

$\displaystyle \mbox{$\cal C$}$ $\displaystyle =\{Y \Delta T(Y) \; \vert \; Y \in$ $\displaystyle \mbox{$\cal A$}$ $\displaystyle \}$

of coboundaries. How big is the cohomology group

$\displaystyle \mbox{$\cal H$}$ $\displaystyle (T,\bf {Z}_2)=$ $\displaystyle \mbox{$\cal A$}$ $\displaystyle /$ $\displaystyle \mbox{$\cal C$}$ $\displaystyle \; ?$

We call the problem the cohomology problem for measurable sets. An other problem is to decide whether a given measurable set is a coboundary or not. We call this problem the coboundary identification problem for measurable sets. We will see in the chapter "Discontinuity and positivity of Lyapunov exponents" that solving this problem is easier than calculating Lyapunov exponents.

For finite periodic dynamical systems, the problem can be solved: assume $ X$ is a finite set and $\mbox{$\cal A$}$ is the set of subsets of $ X$ . An ergodic measure preserving map $T:X \mapsto X$ is just a cyclic permutation of $ X$ . The group $\mbox{$\cal A$}$ consists of all subsets of $ X$ and has $2^{\vert X\vert}$ elements. It is quite easy to see that $\mbox{$\cal C$}$ consists exactly of the sets with even cardinality. In this case $\mbox{$\cal H$}$ $(T,\bf {Z}_2)=$ $\mbox{$\cal A$}$ $ /$ $\mbox{$\cal C$}$ is isomorphic to $\bf {Z}_2$ .

There are a lot of other related unsolved questions. For example, there is a conjecture of Kirillov: assume two automorphism of a fixed probability space $ (X,m)$ have the same measurable sets as coboundaries. Are they conjugate?

The cohomology problem for measurable sets can be generalized as a question in group theory: let $\mbox{$\cal G$}$ be an arbitrary abelian group and $ T$ a group automorphism. What is the group $\mbox{$\cal H$}$ $ ($ $\mbox{$\cal G$}$ $ ,T)=$ $\mbox{$\cal G$}$ $ /$ $\mbox{$\cal C$}$ , where

$\displaystyle \mbox{$\cal C$}$ $\displaystyle =\{g(T)g^{-1} \; \vert \; g \in$ $\displaystyle \mbox{$\cal G$}$ $\displaystyle \} \; ?$

Let $ (X,T,m)$ be a $\bf {Z}^d$ action. A generalization of the cohomology problem for sets is to find the moduli space of all zero curvature $\bf {Z}_2$ fields. A gauge field $Y=(Y_1,Y_2, \dots, Y_d)$ is given by $ d$ measurable sets $ Y_i$ . The space $\mbox{$\cal E$}$ of all fields is the d-fold direct product of the group of all measurable sets. The curvature of such a gauge field $ Y$ is

$\displaystyle F_{ij}(Y)= Y_i \Delta Y_j(T_i) \Delta Y_i(T_j) \Delta Y_j \; .$

A field

has zero curvature if $F_{ij}(Y)=\emptyset$ for all $ i,j$

. There is a subgroup of $\mbox{$\cal E$}$ called the group of gradients

$\displaystyle \mbox{$\cal C$}$ $\displaystyle =\{ Y \in$ $\displaystyle \mbox{$\cal E$}$ $\displaystyle \; \vert \; \exists Z, \; \forall i=1, \dots d, Y_i=Z \Delta Z(T_i) \} \; .$

On $\mbox{$\cal E$}$ are defined the Gauge transformations

$\displaystyle Y_i \rightarrow Y_i \Delta Z(T_i) \Delta Z \; .$

The gradients are just the fields which can be gauged to the identity.
The question is to find the moduli space of all zero curvature fields. In other words, we want to find the group

$\displaystyle \mbox{$\cal H$}$ $\displaystyle =$ $\displaystyle \mbox{$\cal E$}$ $\displaystyle /$ $\displaystyle \mbox{$\cal C$}$ $\displaystyle \; .$

We have used only the additive group structure of the measurable sets. Taking the ring structure also gives also interesting problems which lead more away from the subject. The problem is however of the same type. As an example we consider a random version of a nonlinear cellular automata recently studied by Bobenko, Bordemann, Gunn and Pinkall [Bbgp 92]. It is an "integrable" system and the evolution can be interpreted as a collision process of soliton like particles. The cellular automata can be described as a $\bf {Z}_2$ -valued field $X_{nm}$ on a two-dimensional lattice $\bf {Z}^2$ satisfying the rule

(1)

$\displaystyle X_{n,m}+X_{n+1,m+1}=X_{n,m+1} X_{n+1,m} + X_{n,m+1} + X_{n+1,m} \pmod{2} \;.$

We translate this automata now into a random setting. Given a $\bf {Z}^2$ dynamical system $ (X,T_1,T_2,m)$

. Assume we have given a measurable set $Y \subset X$ satisfying

(2)

$\displaystyle Y \Delta Y(T_1T_2)=Y(T_1) \cap Y(T_2) \Delta Y(T_1) \Delta Y(T_2) \;.$

Define the random variable $ X(x)=1_Y$

. For almost all $x \in X$ we can write $X_{n,m}=X(T_1^nT_2^m)$ and get from Equation 2 the rule 1. Equation 2 is equivalent to

$\displaystyle Y \Delta Y(T_1T_2)=Y(T_1) \cup Y(T_2) \; .$

The problem is to find nontrivial sets satisfying this equation. Is there for any cellular automata rule

$\displaystyle Y(T_1,T_2)=F(Y,Y(T_1),Y(T_2)) \; ,$

where

is one of the $ 2^8$

possible functions a measurable set which solves it?

An abstract generalization of the cellular automata is obtained as follows. Let $ ($ $\mbox{$\cal R$}$ $,+,\cdot)$ be a commutative ring over the field $\bf {Z}_2$ . Given two automorphism $ T_1,T_2$ of this ring. Use the notation $ T_1(r)=r(T_1)$ for $r \in$ $\mbox{$\cal R$}$ . The question is if there exists a non-zero ring element $r \in$ $\mbox{$\cal R$}$ satisfying

$\displaystyle r + r(T_1 T_2)=r(T_1) \cdot r(T_2) + r(T_1)+r(T_2) \; .$

In this case, every ring element $ T_1^nT_2^m(r)$

describes the cellular automata BBGP.

In the two-dimensional BBGP automata, the time axis is given by the transformation $ T_1 T_2$ and in a natural way, the propagation of particles can't be bigger then the "speed of light" $ 1$ . A natural generalization to three dimensions would be a rule

$\displaystyle Y \Delta Y(T_1T_2T_3)= F(Y(T_i),Y(T_i,T_j)) \;,$

where

should be symmetric with respect to permutations of $ T_i$

and cyclic substitutions

$\displaystyle Y_1 \mapsto Y(T_1T_2) \mapsto Y(T_2) \mapsto Y(T_2T_3) \mapsto Y(T_3) \mapsto Y(T_3T_1) \; .$

We summarize: the cohomology identification problem would be solved if we could calculate the Lyapunov exponents of symplectic cocycles. Cohomology problems over an abstract dynamical system with structure group is $\bf {Z}_2$ lead to interesting questions. The general problem of calculating the cohomology groups $ H^1(T_1,G)$ is an example of a cohomology problem.

Bibliography

Adl 81

M.Adler.
On the Bäcklund Transformation for the Gelfand-Dickey Equations.

Commun. Math. Phys., 80:517-527, 1981

Aha 90

D.Aharonov, U.Elias.
Invariant curves around a parabolic fixed point at infinity.

Ergod. Th. Dynam. Sys., 10:209-229, 1990

Alv 91

L.Alvarez-Gaume.
Integrability in Random Matrix Models.

CERN preprint June, 1991

Alv 91a

L.Alvarez-Gaume, C.Gomez, J.Lacki.
Integrability in random matrix models.

Physics Letters B, 253:56-62, 1991

Akc 65

M.A.Akcoglu,R.V.Chacon.
Generalized eigenvalues of automorphisms.

Proc.Amer.Math.Soc., 16:676-680, 1965

Ang 90

S.Angenent.
Monotone recurrence relations their Birkhoff orbits and topological entropy.

Erg. Th. Dyn., Sys., 10:15-41, 1990

Ang 92

S.Angenent.
A remark on the topological entropy and invariant circles of an area preserving twist map.

In "Twist mappings and their Applications", IMA Volumes in Mathematics, Vol. 44., Eds. R.Mc Gehee, K.Meyer, 1992

Anz 51

H.Anzai.
Ergodic Skew Product Transformations on the Torus.

Osaka Math. J., 3:83-99, 1951.

Are 83

H.Aref.
Integrable, chaotic and turbulent vortex motion in two dimensional flows.

Ann. Rev. Fluid. Mech., 15:345-389, 1983

Arn 88

V.Arnold.
Geometrical methods in the theory of ordinary differential equations.
Springer Grundlehren No. 250, New York, 1988

Arn 89

V.I. Arnold.
Mathematical methods of classical mechanics.

Springer, New York (etc.), 1989

Aub 92

S.Aubry.
The concept of anti-integrability: definition, theorems and applications to the standard map.

In "Twist mappings and their Applications", IMA Volumes in Mathematics, Vol. 44., Eds. R.Mc Gehee, K.Meyer, 1992

Aub 90

S.Aubry, G.Abramovici.
Chaotic Trajectories in the Standard map. The concept of anti-integrability.

Physica D, 43:199-219, 1990

Aub 92

S.Aubry, R.S.MacKay, C.Baesens.
Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models.

Physica D, 56:123-134, 1992

Avr 90

J.Avron, P. van Mouche, B.Simon.
On the measure of the spectrum for the almost Mathieu model.

Commun. Math. Phys., 132:103-118, 1990

Bag 88

L.Baggett.
On circle valued cocycles of an ergodic measure preserving transformation.

Israel J. Math., 61:29-38, 1988.

Bak 84

G.A.Baker, D.Bessis, P.Moussa.
A family of almost periodic Schrödinger operators.

Physica A, 124:61-78, 1984

Ban 88

V.Bangert.
Mather Sets for Twist Maps and Geodesics on Tori.

Dynamics Reported, 1:1-55, 1988

Ban 92

J.Banks, J.Brooks,G.Cairns,G.Davis,P.Stacey.
On Devaney's definition of chaos.

Americ. Math. Month.,99:332-334, 1992

Bar 83

M.F.Barnsley, J.S.Geronimo, A.N. Harrington.
Geometry, electrostatic measure and orthogonal polynomials on Julia sets for polynomials.

Ergodic Th. Dynam. Sys., 3:509-520, 1983

Bar 83a

M.F.Barnsley, J.S.Geronimo, A.N. Harrington.
Infinite-Dimensional Jacobi Matrices associated with Julia sets.

Proc. Amer.Math.Soc, 88:625-630, 1983

Bar 83b

M.F.Barnsley, J.S.Geronimo, A.N. Harrington.
On the Invariant Sets of a Family of Quadratic Maps.

Commun. Math. Phys., 88:479-501, 1983

Bar 85

M.F.Barnsley, J.S.Geronimo, A.N. Harrington.
Almost periodic Jacobi Matrices associated with Julia sets for Polynomials.

Commun. Math. Phys., 99: 303-317, 1985

Bar 88

M.Barnsley.
Fractals everywhere.

Academic Press, London, 1988

Bbgp 92

A.Bobenko,M.Bordemann,C.Gunn,U.Pinkall.
On two integrable cellular automata.

SFB 288 Preprint No.25, Technische Universität Berlin

Bel 85

J.Bellisard.
theory of algebras in solid state physics.

Lecture Notes Phys., 257: 99-156, 1986

Bel 82

J.Bellisard, D.Bessis, P.Moussa.
Chaotic States of Almost Periodic Schrödinger operators.

Phys. Rev. Lett., 49:700-704, 1982

Bel 92

J.Bellisard.
Le papillon de Hofstadter.

, 206 Séminaire Bourbaki 44ème année.

Ben 78

G.Benettin,J.Strelcyn. Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy.

Phys. Review A, 17:, 1978

Ber 86

Yu.M.Berezansky.
The integration of semi-infinite Toda Chain by means of inverse spectral problem.

Reports on mathematical physics, 24:21-47, 1986

Ber 86b

Yu.M.Berezansky, M.I.Gekhtman,M.E.Shmoish.
Integration of some chains of nonlinear difference equations by the method of the inverse spectral problem.

Ukrainian. Math. J., 38:74-78, 1986

Bes 82

D.Bessis, M.L.Mehta, P.Moussa.
Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings.

Lett math. Phys., 6: 123-140, 1982

Bla 84

P.Blanchard.
Complex analytic dynamics on the Riemann sphere.

Bull. (New Ser.) of the Amer. Math. Soc., 11:85-141, 1984

Bob 93

A.Bobenko,N.Kutz,U.Pinkall.
The Discrete Quantum Pendulum.

Physics letters A 177, 399-304, 1993

Bog 88

O.I.Bogoyavlensky.
Five Constructions of Integrable Dynamical Systems.

Acta Appl. Math., 13:227-266, 1988

Bog 90

O.I.Bogoyavlensky.
On Integrable Generalizations of Volterra Systems.

in Analysis, et cetera, ed. by P.Rabinowitz, E.Zehnder, Academic Press, Boston 1990

Bog 76

O.I.Bogoyavlensky.
On Perturbations of the Periodic Toda lattice.

Commun. Math. Phys., 51:201-209, 1976

Bow 75

R.Bowen.
Equilibrium States and the ergodic theory of Anosov diffeomorphisms.

Lect. Notes. Math., 470, Springer 1975

Bow 75

R.Bowen, D.Ruelle.
The Ergodic Theory of Axiom A Flows.

Invent. Math., 29:181-202, 1975

Bra 88

G.Brassard.
Modern Cryptology.

Lecture Notes in Computer Science, 325, Springer 1988

Bra 81

M.Braun.
Mathematical remarks on the van Allen radiation belt: A survey of old and new results.

SIAM Review, 23:61-93, 1981

Bra 81a

M.Braun.
Invariant curves, homoclinic points and ergodicity in area-preserving mappings.

SIAM J.Math.Anal, 12:630-638, 1981

Bro 65

H.Brolin.
Invariant sets under itaration of rational functions.

Arkiv Math., 6: 103-144, 1965

Bro 75

M.Brown, W.D.Neuman.
Proof of the Poincaré-Birkhoff fixed point theorem.

Michigan Math. J. 24: 21-31, 1975

Cap 91

H.Capel, F.Nijhoff, V.Papageorgiou.
Complete integrability of Lagrangian mappings and lattices of KdV type.

Physics Letters A, 155:377-387, 1991

Car 87

N.Caranicolas, Ch.Vozikis.
Chaos in a quartic dynamical model.

Celestial Mechanics, 40:35-49, 1987

Car 87

R.Carmona, S.Kotani.
Inverse Spectral Theory for Random Jacobi Matrices.

Journal Statistical Physics, 46:1091-1114, 1987

Car 90

R.Carmona, J.Lacroix.
Spectral Theory of Random Schrödinger Operators.

Birkhäuser, Boston, 1990

Cel 88

A. Celletti A., L. Chierchia.
Construction of analytic KAM surfaces and effective stability bounds.

Commun. Math. Phys., 118:119-161, 1988

Con 89

G.Contopoulos, B.Barbanis.
Lyapunov characteristic numbers and the structure of phase space.

Astron. Astrophys., 222:329-343, 1989

Con 89

A.Connes.
Compact metric spaces, Fredholm modules, and hyperfiniteness.

Ergod. Th. Dynam. Sys 9:207-220, (1989)

Con 90

A.Connes.
Geometrie non commutative.

InterEditions, Paris 1990

Cor 82

I.P.Cornfeld, S.V.Fomin, Ya.G.Sinai.
Ergodic Theory.

Springer, New York, 1982

Cra 89

W.Craig.
The Trace Formula for Schroedinger Operators on the Line.

Commun. Math. Phys., 126:379-407, 1989

Chu 84

M.T.Chu.
On the global convergence of the Toda lattice for real normal matrices and ists applications to the eigenvalue problem.

SIAM J. Math. Anal. 15 (1), 1984

Chu 84a

M.T.Chu.
The generalized Toda flow, the algorithm and the center manifold theory.

SIAM J. Discrete Math., 5:187-201, 1984

Chu 86

M.T.Chu.
A differential equation approach to the singular value decomposition of bidiagonal matrices.

Linear algebra and its applications, 80:71-79, 1986

Chu 89

V.Chulaevsky.
Lyapunov Exponents and Anderson Localisation for Discrete Schroedinger Operators.

Preprint FIM ETH Zuerich, 1989

Chu 91

V.Chulaevsky, Ya.G.Sinai.
The exponential localization and structure of the spectrum for 1D quasi-periodic discrete Schrödinger operators.

Reviews in Mathem. Phys., 3:241-284, 1991

Cyc 87

H.L.Cycon,R.G.Froese,W.Kirsch,B.Simon.
Schroedinger Operators.

Texts and Monographs in Physics, Springer, 1987

Dam 91

P.Damianou.
About the Toda and Volterra system.

Physics Lett. A, 1991

Dei 78

P.Deift.
Applications of a commutation formula.

Duke Math. J., 45: 267-310, 1978

Dei 91

P.Deift, J.Demmel, L.Li,C.Tomei.
The bidiagonal singular value decomposition and Hamiltonian mechanics.

SIAM J. Numer. Anal., 28:1463, 1991

Deif 80

P.Deift,F.Lund,E.Trubowitz.
Nonlinear Wave Equations and Constrained Harmonic Motion.

Commun. Math. Phys., 74:141-188, 1980

Deif2 80

P.Deift,L.C.Li,C.Tomei.
Toda flows with infinitly many variables.

Journal of Functional analysis, 64:358-402, 1985

Deif3 89

P.Deift,L.C.Li,C.Tomei.
Matrix Factorizations and Integrable Systems.

Commun. Pure Appl. Math., XLII: 443-521, 1989

Deif4 89

P.Deift,L.C.Li,C.Tomei.
Ordinary differential equations and the symmetric eigenvalue problem.

SIAM J. Num. Anal., 20:1-21, 1983

Deif5 89

P.Deift,L.C.Li.
Generalized Affine Lie Algebras and the Solution of a Class of Flows Associated with the QR Eigenvalue Algorithm.

Commun. Pure Appl. Math., XLII: 963-991, 1989

Dei 83

P.Deift,T.Nanda,C.Tomei.
Ordinary differential equations and the symmetric eigenvalue problem.

SIAM J. Numer. Anal., 20:1-21, 1983

Deif5 83

P.Deift, B.Simon.
Almost periodic Schroedinger operators.

Commun. Math. Phys., 90:389-411, 1983

Deim 77

K.Deimling.
Ordinary Differential Equations in Banach Spaces.

Lecture Notes in Math., No. 596, Springer 1977

Del 83

F.Delyon, B.Souillard.
The Rotation Number for Finite Difference Operators and its Properties.

Commun. Math. Phys., 89:415-426, 1983

Den 76

M.Denker,C.Grillenberg,K.Sigmund.
Ergodic Theory on Compact Spaces.

Lecture Notes in Math., No. 527, Springer 1976

Den 82

D.Denning.
Cryptography and data security.

Addison-Wesley Publishing Company, Reading, 1982

Der 93

J.-M. Derrien.
Critères d'ergodicité de cocycles en escalier. Exemples.

316:73-76, 1993

Dev 80

R.Devaney.
Triple Collision in the Planar Isosceles Three Body Problem.

Invent. Math.60:249-267, 1980

Dev 84

R.Devaney.
A piecewise linear model for the zones of instability of an area preserving map.

Physica D10:387-393, 1984

Dev 89

R.Devaney.
An introduction to chaotic dynamical systems.

Redwood City, California (etc.) : Addison-Wesley, 1989

Die 68

J.Dieudonné.
Foundations of Modern Analysis, Tome I.

Academic Press,New York and London 1968

Dix 81

J.Dixmier.
Von Neumann Algebras.

North-Holland mathematical Library, Vol 27, Amsterdam, 1981

Don 88

V.Donnay.
Geodesic flow on the two-sphere, Part I: Positive measure entropy.

Ergod. Th. Dynam. Sys., 8:531-553, 1988

Don 88

V.Donnay.
Geodesic flow on the two-sphere, Part II: Ergodicity.

Lect. Notes. Math., 1342: 112-153, 1988

Don 91

V.Donnay.
Using Integrability to produce Chaos: Billiards with positive Entropy.

Commun. Math. Phys., 141:225-257, 1991

Dua 82

R.Douady.
Application du théorme des tores invariants.

, 1982

Eck 84

J.P. Eckmann, H.Koch, P.Wittwer.
A computer-assisted proof of universality for are-preserving maps.

Momoirs of the American Mathematical Society: 47/289:111-121, 1984

Ehl 82

F.Ehlers, H.Knörrer.
An algebro-geometric interpretation of the Bäcklund transformation for the Korteweg-de Vries equation.

Comment. Math. Helv., 57:1-10, 1982

Eil 47

S.Eilenberg,S. McLane.
Cohomology theory in abstract groups.

Annals of Mathematics, 48:51-78, 1947.

Eli 91

L.H.Eliasson.
Ergodic Skew-Systems on $\bf {T}^d \times SO(3,\bf {R})$ .

Preprint Forschungsinstitut für Mathematik ETH Zürich.

Ere 90

A.Eremenko, M.Lyubich.
The dynamics of analytic transformations.

Leningrad Math. J.,3:563-634, 1990

Fad 86

L.D.Faddeev,L.A.Takhtajan.
Hamiltonian methods in the theory of Solitons.

Springer, 1986

Fel 77

J.Feldmann, C.Moore.
Ergodic equivalence relations, cohomology and von Neumann algebras I,II.

Transactions of the American Mathematican Society, 234: 289-359, 1977

Fer 92

R.Fernández, J.Fröhlich, A.Sokal.
Random Walks, critical phenomena and triviality in Quantum field theory.

Texts and Monographs in Pysics, Springer, Berlin 1992

Fla 74

H.Flaschka.
On the Toda Lattice II.

Progress of Theoretical Physics, 51:703-716, 1974

Fon 90

E.Fontich.
Transversal Homoclinic Points of a Class of Conservative Diffeomorphisms.

J.Diff.Equ., 87:1-27, 1990

For 91

J.Fornaess, N.Sibony.
Random iterations of rational functions.

Ergod. Th. Dynam. Sys., 11:687-708, 1991

Fri 70

N.A.Friedman.
Introduction to Ergodic Theory.

Van Nostrand-Reinhold, Princeton, New York, 1970

Fro 90

J.Fröhlich, T.Spencer, P.Wittwer.
Localization for a Class of One Dimensional Quasi-Periodic Schrödinger Operators.

Commun. Math. Phys., 132:5-25, 1990

Fur 63

H.Furstenberg.
Non-commuting random products.

Trans.Amer.Math.Soc.,108:377-428, 1963

Fur 81

H.Fürstenberg.
Recurrence in Ergodic Theory and Combinatorial Number Theory.
Princeton University Press, Princeton, New Jersey, 1981

Gal 81

L.Galgani, A.Giorgilli,J.M.Strelcyn.
Chaotic Motions and Transition to stochasticity in the classical problem of the heavy rigid body with a fixed point.

Il nuovo cimento, 61 B, No 1 , 1981

Gel 92

V.Gelfreich, V.Lazutkin,C.Simo, M.Tabanov.
Fern-like structures in the wild set of the standard and semistandard maps in $\bf {C}^2$ .
International J. of Bif. and Chaos,2: 353-370, 1992

Gen 90

C. Genecand.
Transversal homoclinic points near elliptic fixed points of area preserving diffeomorphism of the plane.

Diss. ETHZ, 1990

Ger 91

A.Gerasimov, A.Marshakov, A.Mironov, A.Morozov, A.Orlov.
Matrix models of two dimensional gravity and Toda theory.

Nuclear Physics B, 357:565-618, 1991

Ges 93

F.Gesztesy, Z.Zhao.
Critical and subcritical Jacobi operators defined as Friedrichs extensions.

J.Diff. Equ., 103:68-93, 1993

Ghe 92

J-M. Ghez.
Schrödinger Operators generated by substitutions.

, Berlin, 1992

Gol 92

C. Golé.
A new proof of Aubry-Mather's theorem.

Math. Z., 210:441-448, 1992

Gol 92a

C.Golé.
Monotone maps of $\bf {T}^n \times \bf {R}^n$ and their periodic orbits.

In "Twist mappings and their Applications", IMA Volumes in Mathematics, Vol. 44., Eds. R.Mc Gehee, K.Meyer 1992

Gol 92

I.Ya.Goldsheid, E.Sorets.
Lyapunov Exponents of the Schrödinger Equation with Quasi-Periodic Potential on a Strip.

Commun. Math. Phys., 145:507-513, 1992

Gol 92a

I.Ya.Goldsheid, E.Sorets.
Lyapunov Exponents of the Schrödinger Equation with Certain Classes of Ergodic Potentials.

Preprint Nr. 147 Institut für Mathematik Ruhr-Universität Bochum, März 1992

Goo 88

J.Goodman, P.D.Lax.
On dispersive Difference Schemes. I.

Commun. Pure Appl. Math., XLI, p. 591, 1988

Gor 85

D.Goroff.
Hyperbolic sets for twist maps.

Ergod. Th. & Dynam. Sys., 5:337-339, 1985

Gro 90

K.L.Gromov.
Spectral Properties of Unitary Weighted Translation Operators.

Functional Analysis and Applications24:233-235, 1990

Gro 90

D.Gross, A.Migdal.
Nonperturbative two dimensional quantum gravity.

Physical review letters, 64: 127-130, 1990

Gui 87

V.Guillemin.
Some inverse spectral results on Riemannian manifolds.

AMS Lecture, Mimeographed copy of a preprint.

Gut 92

E.Gutkin, N.Simanyi.
Dual polygonal billiards and necklace dynamics.

Commun. Math. Phys., 143:431-449, 1992

Hah 32

H.Hahn.
Reelle Funktionen, 1.Teil, Punktfunktionen.

Akademische Verlagsgesellschaft, Leipzig 1932

Hal 88

G. Hall.
A remark on the multiplicity of monotone periodic orbits.

Ergod.Th. Dyn. Sys., 8:109-118, 1988

Hal 56

P.Halmos.
Lectures on Ergodic Theory.

The mathematical society of Japan, 1956

Hay 87

A.Hayli, T.Dumont, J.M.Ollangier, J.M. Strelcyn.
Quelques rsultats nouveaux sur les billard de Robinik.

J.Phys. A : Math. Gen.: 2337-3249, 1987

Hen 64

M.Henon, C.Heiles.
The applicability of the third integral of motion some numerical experiments.

Astronomical Journal, 69:73-79, 1964

Hen 83

M.Henon, J.Wisdom.
The Benettin-Strelcyn oval billiard revisited.

Physica D8:157-169, 1983

Her 81

M.R.Herman.
Construction d'un difféomorphisme minimal d'entropie topologique non nulle.

Ergodic Theory & Dynamical Systems, 1:65-76, 1981

Her 83

M.Herman.
Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2.

Commun.Math.Helv., 58:453-502, 1983

Her 83a

M.Herman.
Sur les courbes invariantes par les difféomorphismes de l'anneau.

, 103-104, Société mathématique de france 1983

Hil 02

D.Hilbert.
Mathematische Probleme.

Gesammelte Abhandlungen III, Berlin 1935

Hof 93

A.Hof.
Some remarks on discrete aperiodic Schrödinger operators.

Journal of Stat. Physics72:1353-1374, 1993

Hua 82

L.K. Hua.
Introduction to Number theory.

Springer, Berlin, 1982

Joh 82

R.Johnson, J.Moser.
The Rotation Number for Almost Periodic Potentials.

Commun. Math. Phys., 84:403-438, 1982

Joh 84

R.A.Johnson.
Lyapunov numbers for the almost periodic Schroedinger equation.

Illinois J. Math., 28:397-419, 1984

Joh 86

R.A.Johnson.
Exponential Dichotomy, Rotation Number and Linear Differential Operators with Bounded Coefficients.

J. Differential Equations, 61:54-78, 1986

Joh 87

R.Johnson, K.Palmer, G.Sell.
Ergodic properties of linear dynamical systems.

SIAM J. Math. Anal., 18:1-33, 1987

Kac 75

M.Kac,P. van Moerbeke.
A complete solution of the periodic Toda problem.

Proc. Nat. Acad. Sci. USA, 72, No.8, p.2879-2880, 1975

Kam 93

S.Kamvissis.
On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity.

Commun. Math. Phys., 153:479-519, 1993

Kat 80

A. Katok. Lyapunov Exponents, entropy, and periodic orbits for diffeomorphisms.
Publ.Math. IHES, 51:137-172, 1980

Kat 82

A. Katok.
Some remarks on Birkhoff and Mather twist map theorems.

Ergod. Th. & Dynam. Sys., 2:185-194, 1982

Kat 86

A.Katok, J.-M.Strelcyn. Invariant manifolds, Entropy and Billards : Smooth maps with Singularities.
Lecture Notes in Math., 1222, Springer 1986

Kif 82

Y.Kifer.
Perturbations of Random Matrix Products.

Z.Wahrsch. theorie verw. Gebiete, 61:83-95, 1982

Kir 67

A.A. Kirillov.
Dynamical systems, factors and representations of groups.

Russian Math. Surveys, 22:63-75, 1967

Kni 2

O.Knill.
Positive Lyapunov exponents for a dense set of bounded measurable cocycles.

Ergod. Th. Dynam. Sys., 12, 319-331, 1992

Kni 1

O.Knill.
The upper Lyapunov exponents of cocycles: Discontinuity and the problem of positivity.

Lect. Notes Math., 1486, Springer 1991

Kni 3

O.Knill.
Isospectral Deformations of Random Jacobi Operators.

Commun. Math. Phys., 151:403-426, 1993

Kni 4

O.Knill.
Factorisation of Random Jacobi Operators and Bäcklund transformations.

Commun. Math. Phys., 151:589-605, 1993

Kni 5

O.Knill.
An analytic map containing the standard map family.

Submitted to Nonlinearity

Kni 6

O.Knill.
Embedding abstract dynamical systems in monotone twist maps.

To be submitted

Koo 86

H.Kook, J.D.Meiss.
Periodic orbits for reversible, symplectic mappings.

Physica D, 35:65-86, 1986

Kos 79

B.Kostant.
The solution to a generalized Toda lattice and representation theory.

Adv. in Math., 34:195-338, 1979

Kot 89

S.Kotani.
Jacobi matrices with random potentials taking finitely many values.

Rev. in Mathematical Phys., 1:129-133, 1989

Kot 88

S.Kotani, B.Simon.
Stochastic Schroedinger Operators and Jacobi Matrices on the Strip.

Commun. Math. Phys, 119:403-429, 1988

Lak 81

V.Lakshmikantham, S.Leela.
Nonlinear Differential equations in abstract spaces.
Pergamon Press, Oxford, 1981

Lan 68

O.Lanford III.
The classical mechanics of one-dimensional systems of infinitly many particles.

Commun. Math. Phys.9:176-191, 1968

Lan 75

O.Lanford III.
Time evolution of large classical systems.

Dynamical Systems: Theory and Applications, Springer Lecture notes of Physics,38:1-111, 1975

Lan 82

O.Lanford III.
A computer-assisted proof of the Feigenbaum conjectures.

Bulletin of the American Mathematical Society, 6:427-434, 1982

Lan 83

O.Lanford III.
Introduction to the mathematical theory of dynamical systems.

Les Houches Session XXXVI, Editors G.Iooss, R.Helleman, R.Stora, North-Holland Publishing Company, Amsterdam 1983

Lan 85

O.Lanford III.
Introduction to hyperbolic sets.
in "Regular and Chaotic Motion in Dynamical systems"

Eds. G.Velo, A.S.Wightman, Nato ASI Ser. B, 118:73-102, 1985

Lan 92

O.Lanford III.
Lectures on dynamical systems.

Notes from lectures held at ETH, 1990-1992.

Las 90

J.Laskar.
The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones.

Icarus, 88:266-291, 1990

Las 93

Y.Last.
A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants.

Commun. Math. Phys., 151:183-192, 1993

Las 93

Y.Last.
Zero measure spectrum for the almost Mathieu operator.

Technion preprint March, 1993

Laz 84

V.F.Lazutkin, D.Ya. Terman.
Percival Variational Principle for
Invariant Measures and Commensurate-Incommensurate Phase Transitions in One Dimensional Chains.

Commun. Math. Phys, 94:511-522, 1984

Led 82

F.Ledrappier.
Quelques Proprietés des Exposants Caracteristiques.

Lecture Notes in Math., No. 1097, Springer 1982

Led 84

F.Ledrappier.
Positivity of the exponents for stationary sequences of matrices.

Lecture Notes in Math., No. 1186,Springer 1984

Led 91

F.Ledrappier, L.-S. Young.
Stability of Lyapunov exponents.

Ergod. Th. Dynam. Syst, 11:469-484, 1991

Lem 89

M.Lemanczyk.
On the Weak Isomorphism of Stricly Ergodic Homeomorphisms.

Monatsh. Math., 108:39-46, 1989

Li 87

L.C.Li.
Long time behaviour of an infinite particle system.

Commun. Math.Phys., 110:617-623, 1987

Liu 61

L.A. Liusternik, V.J.Sobolev.
Elements of Functional analysis.

Frederick Ungar Publishing Company, New York, 1961

Liv 72

A.Livsic.
Cohomology of dynamical systems.

Izv. Akad. Nauk SSSR, 36:1278-1301, 1972

Lju 83

M.Ju. Ljubich.
Entropy properties of rational endomorphisms of the Riemann sphere.

Ergod. Th. Dynam.Sys., 3:351-385, 1983

Mac 63

G. W. Mackey.
Ergodic theory, group theory and differential geometry
Proc. Nat. Acad. Sci. US, 50:1184-1191, 1963

Mac 83

R.S Mackay.
A renormalisation approach to invariant circles in area-preserving maps.

Physica D, 7:283-300, 1983

Mac 92

R.S. Mackay, J.D. Meiss.
Cantori for symplectic maps near the anti-integrable limit.

Nonlinearity, 5:149-160, 1992

Man 81

R.Mané.
A proof of Pesin's formula.

Erg. Th. Dyn. Syst., 1: 95-102 (1981)

Man 83

R.Mané.
Oseledec's theorem from a generic viewpoint.

Proceedings of the International Congress of Mathematicians, Warschau, 1983

Man 83a

R.Mané.
The Lyapunov exponents of generic area preserving diffeomorphism.

Unpublished

Man 85

R.Mané.
On the Bernoulli property for rational maps.

Ergod. Th. Dynam.Sys., 5:71-88, 1985

Man 77

Yu. I. Manin.
A course in mathematical Logic.

Graduated texts in Mathematics 53, New York, Springer, 1977

Mar 84

C.Marchioro, M.Pulvirenti.
Vortex methods in two dimensional fluid dynamics.

Lecure Notes in Physics 203, Berlin, 1984

Mar 91

E.J.Martinec.
On the Origin of Integrability in Matrix Models.

Commun. Math. Phys., 138:437-449, 1991

Mat 68

J.Mather.
Characterization of Anosov Diffeomorphisms.

Indag. Math., 30:479-483, 1968

Mat 82

J.Mather.
Existence of quasi-periodic orbits for twist homeomorphism of the annulus.

Topology, 21:457-467, 1982

Mat 84

J.Mather.
Amount of rotation about a point and the Morse Index.

Commun. Math. Phys., 94:141-153, 1984

McK 78

H.P. McKean.
Integrable systems and algebraic curves.

Lecture Notes in Math., No. 755, Springer 1978

McK 76

H.P.McKean, E.Trubowitz.
Hill's Operator and Hyperelliptic Function Theory in the Presence of Infinitely many Branch Points.

Commun. Pure Appl. Math., XXIX, 143-226, 1976

McK 89

H.P.McKean.
Is there an infinite dimensional algebraic geometry? Hints from KdV.

Proc. of Symposia in Pure Mathematics, 49:27-38, 1989

Mee 90

R.W.Meester.
Two models of dependent percolation.

Dissertation Technische Universiteit Delft, 1990

Mei 92

J.D.Meiss.
Symplectic maps, variational principles and transport
Reviews of Modern Physics64:795-848, 1992

Mer 85

K.D.Merrill.
Cohomology of step functions under irrational rotations.

Israel J. Math., 52:320-340, 1985.

Mil 91

J.Milnor.
Dynamics in one complex variable.

Introductory Lectures, SUNY, 1991

Moo 78

C.Moore,K.Schmidt.
Coboundaries and homomorphisms for non-singular actions and a problem of H.Helson.

Proc. London Math.Soc., 40:443-475, 1980

Mor 76

P.v.Moerbeke.
The spectrum of Jacobi Matrices.

Invent. Math., 37:45-81, 1976

Mor 78

P.v.Moerbeke.
About isospectral deformations of discrete Laplacians.

Lecture Notes in Mathematics, No. 755, Springer 1978

Mos 73

J. Moser.
Stable and random Motion in dynamical systems.

Princeton University Press, Princeton, 1973

Mos 75

J.Moser.
Three Integrable Hamiltonian Systems Connected with Isospectral Deformations.

Adv. in Math., 16:197-220, 1975

Mos 75

J.Moser.
Finitly many mass points on the Line under the influence of an exponential potential. An integrable system.

Lecture Notes in Physics, 38, 1975

Mos 84

J.Moser.
Drei Kapitel über integrable Hamiltonsche Systeme.

ETH lecture, Summer 1984

Mos 86

J. Moser.
Recent Developments in the theory of Hamiltonian systems.

SIAM Review, 28:459-485, 1986

Mos 87

J. Moser.
On the construction of invariant curves and Mather sets via a regularized variational principle.

in P.H.Rabinowitz et al (eds.) "Periodic Solutions of Hamiltonian Systems and Related Topics":221-234, 1987

Nak 90

Y.Nakamura.
Moduli Space of SU(2) Monopoles and Complex Cyclic-Toda Hierarchy.

Commun. Math.Phys, 128:509-520, 1990

Ner 88

M.G.Nerurkar.
On the construction of smooth ergodic skew-products.

Ergod. Th. Dynam. Syst., 8:311-326, 1988

Neu 32

J. von Neumann.
Zur Operatorenmethode in der Mechanik.

Ann. Math. 33: 587-642, 1932

Orn 70

D.S.Ornstein.
Factors of Bernoulli shifts are Bernoulli shifts.

Adv. Math., 5:349-364, 1970

Pag 89

E. Le Page.
Regularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications.

, 25:109-142, 1989

Par 86

M.Paramio, J.Sesma.
A continued fraction Procedure to determine invariant directions in twist mappings.

Physics Letters A, 117:333-336, 1986

Par 80

W. Parry.
Topics in ergodic theory.

Cambridge University Press, 1980

Part 77

K.R.Parthasarathy, K.Schmidt.
On the cohomology of a hyperfinite action.

mboxMonatsh. Math., 84:37-48, 1977

Per 90

A.M.Perelomov.
Integrable Systems of classical Mechanics and Lie Algebras I.

Birkhäuser, Boston 1990

Per 80

I.C. Percival.
Variational principle for invariant tori and cantori.

Amer. Instit. of Physics Conf. Proc., 57:310-320, 1980

Pes 77

Ya.B.Pesin.
Characteristic Lyapunov Exponents and Smooth Ergodic Theory.

Russian Math. Surveys, 32:55-114, 1977

Pet 83

Peterson K.:
Ergodic theory.

Cambridge Studies in advanced mathematics 2, Cambridge University press, 1983

Prz 82

F.Przytycki.
Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour.

Ergod. Th. Dynam.Sys, 2:439-463, 1982

Qui 91

G.Quispel,H.Capel,V.Papageorgiou,F.Nijhoff.
Integrable mappings derived from soliton equations.

Physica A, 173:243-266, 1991

Rag 78

M.S.Raghunathan.
A proof of Oseledec's Multiplicative Ergodic Theorem.

Israel Journal of Mathematics, 32:356-362,1978

Ree 80

M.Reed, B.Simon.
Methods of modern mathematical physics I, Functional analysis.

Academic Press, Orlando, 1980

Rob 83

M.Robnik.
Classical dynamics of a family of billiards with analytic boundaries.

J.Phys.A : Math. Gen., 16:3971-3986, 1983

Rud 62

W.Rudin.
Fourier Analysis on groups.

Interscience tracts in pure and applied mathematics 12, Interscience Publishers, 1962

Rue 78

D.Ruelle.
Thermodynamic Formalism.

Addison-Wesley Publishing Company, London, 1978

Rue 79

D.Ruelle.
Ergodic theory of differentiable dynamical systems.

Publ. math. de l'IHES, 50:27-58, 1979

Rue 82

D.Ruelle.
Repellers for real analytic maps.

Ergod. Th. Dynam. Sys., 2:99-107, 1982

Rue 85

D.Ruelle.
Rotation numbers for diffeomorphisms and flows.

Ann. de l'Inst. Henri Poincare, Physique theoretique, 42:109-115, 1985

Rue 79a

D.Ruelle.
Analycity Properties of the Characteristic Exponents of Random Matrix Products.

Advances in Mathematics, 32:68-80, 1979

Rue 82

D.Ruelle.
The Obsessions of Time.

Commun. Math. Phys., 85:3-5, 1982

Rue 87

D.Ruelle.
Chaotic evolution and strange attractors.

Cambridge university press, Cambridge 1997

Rue 90

D.Ruelle.
Deterministic chaos: the science and the fiction.

Proc. R.Soc. Lond. A, 427:241-248, 1990

Sac 78

R.Sacker, G.Sell.
A spectral theory for linear differential systems.

J.Diff.Equ., 27:320-358, 1978

Sal 89

D. Salamon, E. Zehnder.
KAM theory in configuration space.

Commun. Math. Helv., 64: 84-132, 1989

San 86

V.S.Sanders.
An invitation to von Neumann algebras.

Springer, New York, 1986

Sch 87

R.Schmid.
Infinite dimensional Hamiltonian systems.

Bibliopolis, edizioni di filosofia e scienze, Napoli, 1987

Sch 89

K. Schmidt.
Algebraic ideas in ergodic theory.

Regional conference Series in mathematics No. 76, AMS Providence 1989

Sie 69

C.L.Siegel.
Topics in complex function theory Vol 2.

Wiley-Interscience, New York, 1969

Sha 90

A.B.Shabat, R.I.Yamilov.
Symmetries of nonlinear chains.

Leningrad Math. J., 2:377-400, 1990

Sim 84

B.Simon.
Fifteen Problems in Mathematical Physics.

Perspectives in mathematics, Anniversary of Oberwolfach 1984. Birkhäuser, Basel, 1984

Sim 83

B.Simon.
Kotani Theory for One Dimensional Stochastic Jacobi Matrices.

Commun. Math. Phys., 89:227-234, 1983

Sim 90

B.Simon.
Absence of Ballistic Motion.

Commun. Math. Phys., 134:209-212, 1990

Sin 76

Ya.G.Sinai.
Introduction to ergodic theory

Princeton university press, 1976

Sor 91

E.Sorets, T.Spencer.
Positive Lyapunov Exponents for Schrödinger Operators with Quasi-Periodic Potentials.

Commun. Math. Phys., 142: 543-566, 1991

Spe 86

T.Spencer.
Some rigorous results for random and quasiperiodic potentials.

Phys. A, 140:70-77, 1986

Sta 88

J.Stark.
An exhaustive criterion for the non-existence of invariant cirles for area-preserving Twist maps.

Comm. Math.Phys.117:177-189, 1988

Ste 71

A.M.Stepin.
Cohomologies of automorphism groups of a Lebesgue space.

Functional Anal. Appl., 5:167-168, 1971

Str 89

J.-M.Strelcyn.
The Coexistence Problem for conservative dynamical systems:A review.

Colloquium Mathematicum,LXII:331-345, 1991

Sur 89

Y.B.Suris. Integrable mappings of the standard type.

Funct. Analy. Appl., 23:74-76, (1989)

Sur 90

Y.Suris.
Discrete time generalized Toda lattices: complete integrability and relation with relativistic Toda lattices.

Physics Letters A, 145:113-119, 1990

Sur 91

Y.Suris.
Generalized Toda chains in discrete time.

Leningrad Math. J., 2:339-352, 1991

Sym 80

W.Symes.
Systems of Toda Type, Inverse Spectral Problems, and Representation Theory.

Invent. Math., 59, 13-51, 1980

Sym 82

W.Symes.
The QR algorithm and scattering for the finite nonperiodic Toda lattice.

Physica D, 4:275-280, 1982

Tab 93

S.Tabachnikov.
Commuting dual billiard maps.

Preprint FIM Febr. 1993

Tod 80

H.Toda.
Theory of nonlinear lattices .

Springer. 1981

Tsu 58

M. Tsuji.
Potential theory in modern function theory.

Chelsea publishing company, New York, 1958

Vee 69

W.Veech.
Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2.

Trans.Amer.Math.Soc., 140:1-33, 1969

Ven 89

S. Venakides.
The Continuum Limit of Theta Functions.

Commun. Pure Appl. Math., 42:711-728, 1989

Viv 87

F.Vivaldi, A.Shaidenko.
Global Stability of a class of discontinuous dual billiards.

Commun. Math. Phys., 110:625-640, 1987

Wed 75

M.Wedati, M.Toda.
Bäcklund transformation for the exponential lattice.

J. Phys. Soc. Japan, 39:1196, 1975

Wal 82

P.Walters.
An introduction to ergodic theory.

Graduate texts in mathematics 79, New York, Springer, 1982

Wis 87

J.Wishdom.
Chaotic behaviour in the solar system.

in "Dynamical chaos", Proceedings of a Royal Society discussion Meeting, Eds. M.Berry, I.Percival, N.Weiss, Princeton, 1987

Woj 85

M.Wojtkowski.
Invariant families of cones and Lyapunov exponents.

Ergodic Theory & Dynamical Systems, 5:145-161, 1985.

Woj 86

M.Wojtkowski.
Principles for the Design of Billiards with Nonvanishing Lyapunov Exponents.

Commun. Math. Phys., 105:391-414,1986.

Wol 91

Wolfram Research.
Mathematica Version 2.0.

Illinois 1991

You 86

L.-S.Young.
Random perturbations of matrix cocycles.

Ergodic Theory & Dynamical Systems, 6:627-637, 1986.

You 92

L.-S.Young.
Some open sets of nonuniformly hyperbolic cocycles.

Ergod.Th. Dynam.Sys 13:409-415, 1993

You 93

L.-S.Young.
Lyapunov Exponents for some quasi periodic cocycles.

Preprint University of Arizona, 1993

Zeh 73

E. Zehnder.
Homoclinic points near elliptic fixed points.

Commun. Pure Appol. Math., 26:131-182, 1973

Zig 80

S.L.Ziglin.
Nonintegrability of a problem on the motion of four point vortices.

Soviet math.Dokl. 21:296-299, 1980

Zha 93

X-H.Zhao, K-H.Kwek,J-B.Li, K-L.Huang
Chaotic and resonant streamlines in the ABC flow
SIAM J. Appl. Math, 53:71-77, 1993

Oliver Knill

$\displaystyle \dot{x}$	$\displaystyle =$	$\displaystyle A \sin(z) + C \cos(y)$
$\displaystyle \dot{y}$	$\displaystyle =$	$\displaystyle B \sin(x) + A \cos(z)$
$\displaystyle \dot{z}$	$\displaystyle =$	$\displaystyle C = \sin(y) + B \cos(x) \; ,$