Codimension one structurally stable chain classes

TL;DR

This paper proves that for codimension one cases, any structurally stable chain class of a diffeomorphism is necessarily hyperbolic, extending the stability conjecture to individual chain classes under specific conditions.

Contribution

It establishes that codimension one structurally stable chain classes are hyperbolic, addressing a subtle case of the stability conjecture for individual chain classes.

Findings

Structurally stable chain classes of index 1 or dim M-1 are hyperbolic.

The proof handles non-locally maximal chain classes and indirect continuation of periodic points.

Extends the stability conjecture to a new class of dynamical systems.

Abstract

The well known stability conjecture of Palis and Smale states that if a diffeomorphism is structurally stable then the chain recurrent set is hyperbolic. It is natural to ask if this type of results is true for an individual chain class, that is, whether or not every structurally stable chain class is hyperbolic. Regarding the notion of structural stability, there is a subtle difference between the case of a whole system and the case of an individual chain class. The later case is more delicate and contains additional difficulties. In this paper we prove a result of this type for the later case, with an additional assumption of codimension 1. Precisely, let $f$ be a diffeomorphism of a closed manifold $M$ and $p$ be a hyperbolic periodic point of $f$ of index 1 or $\dim M-1$. We prove if the chain class of $p$ is structurally stable then it is hyperbolic. Since the chain class of $p$ is not assumed in advance to be locally maximal, and since the counterpart of it for the perturbation $g$ is defined not canonically but indirectly through the continuation $p_g$ of $p$, the proof is quite delicate.