Constraints On Dynamics Preserving Certain Hyperbolic Sets

TL;DR

This paper explores how the topology of certain hyperbolic sets imposes constraints on the dynamics of the ambient system, revealing invariance properties under different diffeomorphisms.

Contribution

It establishes conditions under which hyperbolic sets maintain their topological and dynamical properties across different diffeomorphisms.

Findings

Hyperbolic attractors of codimension 1 are unions of similar attractors or repellers under any hyperbolic diffeomorphism.

Hyperbolic sets on surfaces are locally maximal for all hyperbolic diffeomorphisms.

Topology of hyperbolic sets constrains the possible dynamics in the ambient space.

Abstract

We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First if a set is a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, then it is a union of transitive, expanding attractors (or contracting repellers) of codimension 1 for any diffeomorphism such that it is hyperbolic. Secondly, if a set is a nonwandering, locally maximal, compact hyperbolic set for a surface diffeomorphism, then it is locally maximal for any diffeomorphism for which it is hyperbolic.