Multidimensional $C^0$ transversality and the shadowing property for Axiom A diffeomorphisms
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.06588
- arXiv:
- arXiv:2407.06588
- Bibcode:
- 2024arXiv240706588M
- Keywords:
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- Mathematics - Dynamical Systems;
- 37D20;
- 37C50