Abstract
We introduce a Hamiltonian system with many degrees of freedom for which the nonvanishing of (some) Lyapunov exponents almost everywhere can be established analytically.
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[A-S] Anosov, D. V., Sinai, Ya. G.: Certain smooth ergodic systems. Russ. Math. Surv.22, 103–167 (1967)
[B-B] Ballmann, W., Brin, M.: On the ergodicity of geodesis flows. Erg. Th. Dyn. Syst.2, 311–315 (1982)
[B1] Bunimovich, L. A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys.65, 295–312 (1979)
[B2] Bunimovich, L. A.: Many-dimensional nowhere dispersing billiards with chaotic behavior. Physica D33, 58–64 (1988)
[Bu] Burns, K.: Hyperbolic behavior of geodesic flows on manifolds with no focal points. Erg. Th. Dyn. Syst. 3, 1–12 (1983)
[B-E] Burton, R., Easton, R. W.: Ergodic properties of linked twist mappings. Lecture Notes in Mathematics, vol.819, pp. 35–49. Berlin, Heidelberg, New York: Springer 1980
[B-G] Burns, K., Gerber, M. G.: Real analytic Bernoulli geodesic flows onS 2. Erg. Th. Dyn. Syst. (to appear)
[Ch-S] Chernov, N. I., Sinai, Ya. G.: Entropy of the gas of hard spheres with respect to the group of time space translations. Proceedings of the I. G. Petrovsky Seminar Vol.8, 218–238 (1982)
[C-F-S] Cornfeld, I. P., Fomin, S. V., Sinai, Ya. G.: Ergodic Theory. Berlin, Heidelberg, New York: Springer 1982
[De1] Devaney, R. L.: A piecewise linear model for the zones of instability of an area presenting map. PhysicaD10, 383–393 (1984)
[De2] Devaney, R. L.: Linked twist mappings are almost Anosov. Lecture Notes in Mathematics, vol.819, pp 35–49. Berlin, Heidelberg, New York: Springer 1980
[Do1] Donnay, V. J.: Convex billiards with positive entropy. (In preparation)
[Do2] Donnay, V. J.: Geodesic flow on the two-sphere. Part I: Positive measure entropy. Erg. Th. Dyn. Syst.8, 531–553 (1988)
[D-L] Donnay, V. J., Liverani, C.: Ergodic properties of particle motion in potential fields. (Preprint 1989)
[G] Gerber, M.: Conditional stability and real analytic pseudo-Anosov maps. AMS Memoirs, No.321, (1985)
[H-D-M-S] Hayli, A., Dumont, T., Moulin-Ollagnier, J., Strelcyn, J.-M.: Quelques resultats nouveaux sur les billiards de Robnik. J. Phys. A: Math. Gen.20, 3237–3249 (1987)
[Kn] Knauf, A.: Ergodic and topological properties of Coulombic periodic potentials. Commun. Math. Phys.110, 89–112 (1987)
[K-S] Katok, A. Strelcyn, J.-M., with collaboration of Ledrappier, F., Przytycki, F.: Invariant manifolds, entropy and billiards; smooth maps with singularities. Lecture Notes in Mathematics, vol1222. Berlin, Heidelberg, New York: Springer 1986
[L-M] Lehtihet, H. E., Miller, B. N.: Numerical study of a billiard in a gravitational field. PhysicaD21, 94–104 (1986)
[M] Markarian, R.: Billiards with Pesin region of measure one. Commun. Math. Phys.118, 87–97 (1988)
[O] Oseledets, V. I.: The multiplicative ergodic theorem. The Lyapunov characteristic numbers of a dynamical system. Trans. Mosc. Math. Soc.19, 197–231 (1968)
[P] Pesin, Ya. B.: Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surv.32, 55–114 (1977)
[Pr1] Przytycki, F.: Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviors. Ergod. Th. Dyn. Syst.2, 439–463 (1982)
[Pr2] Przytycki, F.: Ergodicity of toral linked twist mappings. Ann. Sci. Ecole Norm Sup.16, 345–354 (1983)
[Ro] Robnik, M.: Classical dynamics of a family of billiards with analytic boundaries. J. Phys. A,16, 3971–3986 (1983)
[Ru] Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math. IHES50, 27–58 (1979)
[S1] Sinai, Ya. G.: Development of Krylov's ideas. Afterword to N. S. Krylov, Works on the foundations of statistical physics. Princeton, NJ: Princeton University Press 1979
[S2] Sinai, Ya. G.: Dynamical Systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv.25, 137–189 (1970)
[W1] Wojtkowski, M. P.: A model problem with the coexistence of stochastic and integrable behavior. Commun. Math. Phys.80, 453–464 (1981)
[W2] Wojtkowski, M. P.: On the ergodic properties of piecewise linear perturbations of the twist map. Ergod. Th. Dyn. Syst.2, 525–542 (1982)
[W3] Wojtkowski, M. P.: Measure theoretic entropy of the system of hard spheres. Ergod. Th. Dyn. Syst.8, 133–153 (1988)
[W4] Wojtkowski, M. P.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys.105, 319–414 (1986)
[W5] Wojtkowski, M. P.: Linked twist mappings have theK-property. Annals of the New York Academy of Sciences Vol.357, Nonlinear Dynamics, 65–76 (1980)
[W6] Wojtkowski, M. P.: Linearly stable orbits in 3 dimensional billiards. (Preprint 1989)
[W7] Wojtkowski, M. P.: Invariant families of cones and Lyapunov exponents. Ergod. Th. Dyn. Syst.5, 145–161 (1985)
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Communicated by T. Spencer
Supported in part by the Sloan Foundation and the NSF Grant DMS-8807077
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Wojtkowski, M.P. A system of one dimensional balls with gravity. Commun.Math. Phys. 126, 507–533 (1990). https://doi.org/10.1007/BF02125698
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DOI: https://doi.org/10.1007/BF02125698