Combinatorics of One-Dimensional Hyperbolic Attractors of Diffeomorphisms of Surfaces

A. Yu. Zhirov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 143-215

Résumé

An algorithmic solution is given to the two following problems. Let $\Lambda _f$ and $\Lambda _g$ be one-dimensional hyperbolic attractors of diffeomorphisms $f\colon M\to M$ and $g\colon N\to N$, where $M$ and $N$ are closed surfaces, either orientable or not. Does there exist a homeomorphism $h\colon U(\Lambda _f)\to V(\Lambda _g)$ of certain neighborhoods of attractors such that $f\circ h=h\circ g$ (the topological conjugacy problem). Given $h>0$, find a representative of each class of topological conjugacy of attractors with a given structure of accessible boundary (boundary type) for which topological entropy is no greater than $h$ (the problem of enumeration of attractors). The solution of these problems is based on the combinatorial method, developed by the author, for describing hyperbolic attractors of surface diffeomorphisms.