Ergodic Theory of Chaotic Dynamical Systems
A dynamical system in this article consists of a map of a manifold to itself or a flow generated by an autonomous system of ordinary differential equations. For definiteness, we shall discuss the discrete-time case, although most things here have their continuous-time versions. We shall not attempt to define “chaos, ” except to mention two of its essential ingredients:
exponential divergence of nearby orbits
lack of predictability.
KeywordsLyapunov Exponent Invariant Measure Ergodic Theory Unstable Manifold Borel Probability Measure
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- [A]Anosov, D., Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Inst. Math. No. 90 1967. Amer. Math. Soc. 1969 (translation), pp. 1–235.Google Scholar
- [BY2]——— and ———, SBR measures for certain Henon maps, to appear in Inventiones (1993).Google Scholar
- [BK]Brin, M. and Katok, A., On local entropy, Geometric Dynamics, Springer Lecture Notes in Mathematics, No. 1007, Springer-Verlag, Berlin, 1983, pp. 30–38.Google Scholar
- [BL]Blokh, A.M. and Lyubich, M. Yu., Measurable dynamics of S-unimodal maps of the interval, preprint, 1990.Google Scholar
- [Bo]Bowen, R., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer Lectures Notes in Mathematics No. 470, Springer-Verlag, Berlin, 1975.Google Scholar
- [C]Collet, P., Ergodic properties of some unimodal mappings of the interval, preprint, 1984.Google Scholar
- [FHY]Fathi, A., Herman, M., and Yoccoz, J.-C., A proof of Pesin’s stable manifold theorem, Springer Lecture Notes in Mathematics. No. 1007, Springer-Verlag, Berlin, 1983, pp. 117–215.Google Scholar
- [KS]——— and Strelcyn, J.M., Invariant Manifolds, Entropy and Billiards;Smooth Maps with Singularities, Springer Lecture Notes in Mathematics No. 1222, Springer-Verlag, Berlin, 1986.Google Scholar
- [Ki3]———, Random Perturbations of Dynamical Systems, Birkhauser, Basel, 1986.Google Scholar
- [M2]———, On the dimension of compact invariant sets of certain nonlinear maps, Springer Lecture Notes in Mathematics No. 898, Springer-Verlag, Berlin, 1981, pp. 230–241 (see erratum).Google Scholar
- [M3]———, A proof of the C 1-stability conjecture, Publ. Math. IHES 66 (1988), 160–210.Google Scholar
- [N]Newhouse, S., Lectures on Dynamical Systems, Birkhauser, Basel, (1980), 1–114.Google Scholar
- [Si4]———(ed.), Dynamicals Systems II, Springer-Verlag, Berlin, 1989.Google Scholar
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