ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 58F15 Turaev, Dmitry, WIAS, Mohrenstr.39, D-10117 Berlin On infinitely degenerate dynamical systems Given an n-dimensional C^{r}-diffeomorphism f with a homo-/heteroclinic cycle C composed of finitely many periodic and connecting orbits one can easily evaluate an effective dimension d of the problem: the maximal possible number of zero Lyapunov exponents for periodic orbits which can be born in a small neighbourhood of C at small C^{r}-smooth perturbations. Theorem. At d < n, for any d-dimensional C^{r}-diffeomorphism g and for any ε > 0 there exists f^{∗} arbitrarily close to f in C^{r}-topology such that f^{∗} has an absorbing domain D (i. e., (f^{∗})^{N}D ⊂ D for some positive N) in a small neighbourhood of C, and in D there is an attractive d-dimensional invariant manifold M such that (f^{∗})^{N}|_{M} is C^{r}-conjugate to a diffeomorphism g^{∗}, ε-close to g in C^{r}-topology.