Indecomposable continua in exponential dynamics-Hausdorff dimension

TL;DR

This paper investigates the Hausdorff dimension of certain invariant sets and indecomposable continua in exponential dynamics, establishing they have dimension one, which enhances understanding of their geometric complexity.

Contribution

It proves that specific invariant sets and indecomposable continua in exponential dynamics have Hausdorff dimension exactly one, providing new insights into their geometric structure.

Findings

Hausdorff dimension of invariant sets is at most 1

Indecomposable continua in exponential dynamics have Hausdorff dimension equal to 1

Results contribute to understanding the geometric complexity of these sets

Abstract

We study some forward invariant sets appearing in the dynamics of the exponential family. We prove that the Hausdorff dimension of the sets under consideration is not larger than $1$. This allows us to prove, as a consequence, a result for some dynamically defined indecomposable continua which appear in the dynamics of the exponential family. We prove that the Hausdorff dimension of these continua is equal to one.