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Measurements of the Kolmogorov entropy

Since every billiard map T is a measure preserving map on a finite probability space, the Kolmogorov entropy tex2html_wrap_inline853 (= metric entropy) is defined and finite. Pesin theory developed in [16] for tex2html_wrap_inline855 -diffeomorphisms and generalized by Katok and Strelcyn [11] to systems with singularities, states that positive metric entropy is equivalent with the fact that an iterate of the map T is measure theoretically conjugated to a Bernoulli shift on a set of positive measure. In the case of a two-dimensional manifold X like in the billiard case, the Ruelle-Pesin-formula


reduces the calculation of the metric entropy tex2html_wrap_inline853 to the determination of the averaged Lyapunov exponent. This formula is also true for maps with singularities satisfying the Katok-Strelcyn conditions in [11, 13]. The billiard maps tex2html_wrap_inline289 studied here satisfy this condition.

Let us mention here also that the entropy tex2html_wrap_inline853 of a monotone twist map is related to a determinant of an associated discrete Schrödinger operator L which is the Hessian of a natural variational problem associated to the dynamics. By the Thouless formula, one has tex2html_wrap_inline871 (see [12]).

Positive Lyapunov exponents are by Wojtkowsky's theorem equivalent to the existence of an invariant cone bundle in the projective tangent bundle. For billiards, a formula of Wojtkowsky [23] gives a geometrical sufficient condition for the existence of an invariant cone bundle or using quadratic forms [14]. For a refinement of these methods see for example [5]. Unfortunately, no other mathematical tool is yet available to prove mathematically positive Lyapunov exponents for billiards. We computed them numerically.

In the case of the billiard map, there are coordinates (see [23]) in which the Jacobian of the billiard map can be put into the form


where tex2html_wrap_inline875 . Here tex2html_wrap_inline877 is the curvature at an impact point, tex2html_wrap_inline583 is the angle of impact and l is the length of the trajectory between two successive impact points. In our numerical calculations, we took this as the Jacobian.

The first graph Fig 3. shows the entropy of the tex2html_wrap_inline287 -billiard for tex2html_wrap_inline885 . The Kolmogorov entropy was measured for 200 values of p between 1 and 2. For each p the Kolmogorov entropy was found by calculating the Lyapunov exponent at each point of a tex2html_wrap_inline897 grid of initial conditions. At each of these 289 points the Lyapunov exponent was estimated as the value of tex2html_wrap_inline899 for n=25,000 and the average over the 289 initial conditions was plotted.

tex2html_wrap971 The next graph Fig. 4. shows the measured entropy of the tex2html_wrap_inline287 -billiard for tex2html_wrap_inline907 (larger values of p introduce unreasonable numerical errors). We computed the Kolmogorov entropy for 200 values of tex2html_wrap_inline911 and 1000 values of tex2html_wrap_inline913 . We again measured each Kolmogorov entropy by determining the average Lyapunov exponent over a 17 x 17 grid of initial conditions. The length of each orbit was n=3,461.

tex2html_wrap973 As expected, the Kolmogorov entropy tends towards 0 for tex2html_wrap_inline921 and for tex2html_wrap_inline923 . For tex2html_wrap_inline885 , we measured a maximum entropy of 0.28 at p=1.23. For tex2html_wrap_inline931 , the entropy peaked around p=4 to p=6 with an entropy around 0.65.

Comparison between measurements of the entropy for orbits of length 3,461 and orbits of length 25,000 gave differences smaller than 0.05. The graphs indicate, that the entropy is piecewise monotone but that there are bumps.

We look now at the behavior of tex2html_wrap_inline945 near p=2. The measurements show that


with tex2html_wrap_inline951 and tex2html_wrap_inline953 . We observe so a phase transition at p=2. The entropy grows with different growth rates for p;SPMgt;2 and p;SPMlt;2.

We show the numerical values of tex2html_wrap_inline961 in Fig. 5. We computed the entropy for 400 different values of p between 1.8 and 2.2 using again a 17 x 17 grid of initial points with each 5000 iterations so that for this run, 578 million billiard iterations had to be computed and for each orbit, the Jacobian tex2html_wrap_inline969 evaluated.

tex2html_wrap975 Next: Discussion and open questions Up: Billiards in the unit Previous: Invariant curves and topological

Oliver Knill, Jul 10 1998