Abstract
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.
The research presented in this paper was partially supported by the Natural Sciences and Engineering Research Council, Canada
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References
D. V. Anosov, Geodesic flows on closed Riemanian manifolds with negative curvature, Proceedings of the Steklov Institute, 90 (1967) AMS, Providence, RI.
M. L. Blank, Metric properties of ε-trajectories of dynamical systems with stochastic behavior, Ergodic Theory and Dynamical Systems, 8 (1988) 365–378.
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphsims, vol. 470, Lecture Notes in Mathematics, Springer-Verlag, New York, 1975.
R. Bowen, ω-limit sets for axiom A diffeomorphisms, JDE, 18 (1975) 333–339.
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.
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS-NSF Regional Conference Series, 38, AMS, Providence, RI, 1978.
J. Cheeger & D. Ebin, Comparison Theorems in Riemannian Geometry, American Elsevier, New York, 1975.
N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J, 23 (1974) 1109–1137.
D. Grobman, Homeomorphisms of systems of differential equations, Dokl. Akad. Nauk 128 (1965) 880 - 881.
J. Hadamard, Sur l’itération et les solutions asymptotiques des equations différentielles, Bull. Soc. Math. France 29 (1901) 224 - 228.
P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
B. Hasselblatt & A. Katok, Introduction to the Modem Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1994.
M. Hirsch, J. Palis, C. Pugh & M. Shub, Neighborhoods of hyperbolic invariant manifolds, Inventiones Math, 9 (1970) 121–134.
M. Hirsch & C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis, Proc. Symp. Pure Math., 14 (1970).
M., Hirsch, C. Pugh & M. Shub, Invariant Manifolds, vol. 583, Lecture Notes in Mathematics, Springer-Verlag, New York, 1977.
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.
T., Kruger & S., Troubetzkoy, Markov partitions and shadowing for nonuniformly hyperbolic systems with singularities, Ergodic Theory and Dynamical Systems, 12 (1992) 487–508.
J., Mather, Characterization of Anosov diffeomorphisms, Indag. Math, 30 (1968) 479–483.
J., Moser, On a Theorem of D. Anosov, JDE, 5 (1969) 411–440.
Z. Nitecki, Differentiable Dynamics, MIT Press, Cambridge, 1971.
M., Pollicott, Pesin Theory, Cambridge University Press, Cambridge, 1992.
C., Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 5 (1977) 425–437.
S. V., Šlačkov, Pseudo-orbit tracing property and structural stability of expanding maps of the interval, Ergodic Theory and Dynamical Systems, 12 (1992) 573–587.
S. Smale, Differentiable Dynamical Systems, Bull. AMS 73 (1967) 747–817.
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Fenichel, N. (1996). Hyperbolicity and Exponential Dichotomy for Dynamical Systems. In: Jones, C.K.R.T., Kirchgraber, U., Walther, HO. (eds) Dynamics Reported. Dynamics Reported. New Series, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79931-0_1
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