Exponential dichotomies, the shadowing lemma and transversal homoclinic points. (English) Zbl 0676.58025

Dyn. Rep. 1, 265-306 (1988).
[For the entire collection see Zbl 0651.00018.]
Using the so-called shadowing lemma, the author proves the famous Smale theorem [S. Smale, Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton, 63-80 (1965; Zbl 0142.411)] concerning the chaotic behaviour of the dynamics in the neighbourhood of the orbit of a transversal homoclinic point.
The approach is quite new. The shadowing lemma employed is established in terms of the theory of exponential dichotomies. More concretely, he has developed the theory of exponential dichotomies for linear difference equations at first. Then every orbit of the diffeomorphisms under consideration is associated just with a suitable linear difference equation. At last the shadowing lemma is applied to prove Smale’s theorem.
Moreover, the definition of a transversal homoclinic point given by the author is shown to be equivalent to the usual one by means of the stable manifold theorem, and the property of having a transversal homoclinic point is verified to persist under perturbation.
Reviewer: J.Andres


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37C75 Stability theory for smooth dynamical systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion