On Invariant Sets of Diffeomorphisms

TL;DR

This paper provides upper bounds for the box dimension of invariant sets under $C^{1}$-diffeomorphisms and mappings with finite Brouwer degree on Riemannian manifolds, advancing understanding of their geometric complexity.

Contribution

It introduces simple upper bounds for the box dimension of invariant sets in the context of $C^{1}$-diffeomorphisms and mappings with finite Brouwer degree on Riemannian manifolds.

Findings

Upper bound for the box dimension of backward invariant sets.

Upper bound for the box dimension of forward invariant sets.

Applicable to $C^{1}$-diffeomorphisms and finite Brouwer degree mappings.

Abstract

We give a simple upper bound for the upper box dimension of a backward invariant set of a $C^{1}$-diffeomorphism of a Riemannian manifold. We also estimate an upper bound for the box dimension of a forward invariant set of a $C^{1}$-mapping with finite Brouwer degree in a Riemannian manifold.