Thesis [PDF].

Diss. ETH Nr. 10189

Spectral, ergodic and cohomological problems in dynamical systems

A dissertation submitted to the

for the degree of
Doctor of Mathematics

presented by
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


[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).


$ \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.


$ \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.

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.

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$ .

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 $ V_n$ 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}];
   {i,Length[S]}];alpha=alpha+1/m,{m+1}]; Show[Graphics[p]] ];

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(a)=ab$ , $ 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 \; , $

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

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 $ T$ 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 $ Y$ 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

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

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

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

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 $ F$ 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 $ F$ 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.


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