Hyperbolicity and Exponential Dichotomy for Dynamical Systems
The aim of this paper is to show that many of the interesting topological consequences of hyperbolicity follow from just one lemma about exponential dichotomy. Our main lemma asserts that there are local stable manifolds and local unstable manifolds associated with a sequence of maps which are close to hyperbolic linear maps, and that certain local stable manifolds and local unstable manifolds have unique points of intersection. Our main lemma also includes detailed estimates for the positions of the local stable and unstable manifolds, and for the behavior of orbits in the local stable and unstable manifolds. This is an exponential dichotomy result because the hypotheses guarantee that orbits diverge either in the forward direction or in the backward direction. In applications the maps represent a given dynamical system, or dynamical systems in a neighbor hood of a given dynamical system, in local coordinates.
KeywordsPeriodic Orbit Periodic Point Unstable Manifold Stable Manifold Exponential Dichotomy
Unable to display preview. Download preview PDF.
- D. V. Anosov, Geodesic flows on closed Riemanian manifolds with negative curvature, Proceedings of the Steklov Institute, 90 (1967) AMS, Providence, RI.Google Scholar
- R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphsims, vol. 470, Lecture Notes in Mathematics, Springer-Verlag, New York, 1975.Google Scholar
- R. Bowen, ω-limit sets for axiom A diffeomorphisms, JDE, 18 (1975) 333–339.Google Scholar
- C. Conley, Hyperbolic sets and shift automorphisms, in Dynamical Systems, Theory and Applications, Lecture Notes in Physics 38, 539–549, Springer-Verlag, New York, 1975.Google Scholar
- B. Hasselblatt & A. Katok, Introduction to the Modem Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1994.Google Scholar
- M. Hirsch & C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis, Proc. Symp. Pure Math., 14 (1970).Google Scholar
- M., Hirsch, C. Pugh & M. Shub, Invariant Manifolds, vol. 583, Lecture Notes in Mathematics, Springer-Verlag, New York, 1977.Google Scholar
- A., Katok, Dynamical systems with hyperbolic structure, in Three papers on Dynamical Systems, J. Szucz éd., AMS Translations Series 2, 116 (1981) 43–95. Original Soviet publication: Izdanie Inst. Mat., Akad. Nauk. Ukrain. SSR, Kiev (1972) 125–211.Google Scholar
- Z. Nitecki, Differentiable Dynamics, MIT Press, Cambridge, 1971.Google Scholar
- M., Pollicott, Pesin Theory, Cambridge University Press, Cambridge, 1992.Google Scholar
- C., Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 5 (1977) 425–437.Google Scholar