Multidimensional $C^0$ transversality and the shadowing property for Axiom A diffeomorphisms

TL;DR

This paper links a topological notion of $C^0$ transversality for Axiom A diffeomorphisms to the shadowing property, showing homological conditions are sufficient and related to $C^0$ transversality in codimension one cases.

Contribution

It establishes a connection between $C^0$ transversality, homological conditions, and the shadowing property for Axiom A diffeomorphisms, extending previous topological and homological insights.

Findings

Homological condition is sufficient for shadowing in Axiom A diffeomorphisms.

$C^0$ transversality implies the homological condition in codimension one cases.

Homological conditions characterize $C^0$ transversality for certain dynamical systems.

Abstract

Petrov and Pilyugin (2015) generalized a notion of $C^0$ transversality of Sakai (1995) using smooth curves. Their definition involves only continuous maps from ${\mathbb R}^n$ to a manifold, which is a purely topological one. They also provided a sufficient condition for the $C^0$ transversality in terms of homological nature. In this paper, we prove that such a homological condition of Axiom A diffeomorphisms is sufficient for enjoying the shadowing property. Moreover, it is proved that the $C^0$ transversality of Axiom A diffeomorphisms with codimension one basic sets implies the homological condition.