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Ergodicity in Hamiltonian systems. (English) Zbl 0824.58033

Jones, Christopher K. R. T. (ed.) et al., Dynamics reported. Expositions in dynamical systems. New series. Vol. 4. Berlin: Springer-Verlag. 130-202 (1995).
The paper deals with the question of openness of ergodic components of (possibly discontinuous) dynamical systems under some hyperbolicity assumptions. The following three paragraphs are quoted from the Introduction (detailed references contained in the original have been omitted).
“In this work, spanning two decades, on the system of colliding balls \(\dots\) Sinai developed a method of proving (local) ergodicity in discontinuous systems with nonuniform hyperbolic behavior. We will refer to it as the Sinai method. It was improved by Sinai and Chernov, and by A. Krámli, N. Simányi and D. Szász. In both papers the discussion is confined to the realm of semidispersing billiards.”
“The purpose of the present paper is to recover the Sinai method as a part of the theory of hyperbolic dynamical systems. In the process we have simplified some of the aspects of the method, and we have revealed its logical structure and limitations.”
“We rely on two developments. The first is the work of Katok and Strelcyn in which they generalized Pesin theory to discontinuous systems. The other is the development of criteria for nonvanishing of Lyapunov exponent in Hamiltonian systems in papers [by the second author]. \(\dots\) Burns and Gerber found a sufficient condition for (local) ergodicity in the smooth case of lowest dimension \(\dots\). It was later generalized by Katok to arbitrary dimension. As a byproduct of our general approach, which is aimed at discontinuous systems, we obtain a similar theorem \(\dots\) and a new proof.”
The paper also contains a final section concerned with 3 applications. The theory developed in the paper applies to the first two of these, billiards in a convex scattering domain and the piecewise linear standard map, but one of the hypotheses is violated in the third, a system of at least 3 falling balls.
In a short Afterword at the end of the paper the authors note a number of recent and related papers.
For the entire collection see [Zbl 0811.00008].

MSC:

37A99 Ergodic theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37D99 Dynamical systems with hyperbolic behavior
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