The dimension of the non-dense orbit set for expanding maps

TL;DR

This paper provides a new proof for the Hausdorff dimension of non-dense orbit sets in expanding maps, linking it to repellers and their dimensions, enhancing understanding of dynamical system complexity.

Contribution

It introduces a novel proof method for the Hausdorff dimension of non-dense orbit sets, utilizing bounds on repellers and establishing dimension equivalences.

Findings

Hausdorff dimension of non-dense orbit sets is characterized

Carathéodory singular dimension equals that of repellers

Provides a new proof based on bounds of repellers

Abstract

In this paper, we give a new proof for the Hausdorff dimension of the non-dense orbit set for expanding maps. This proof is based on the sharp lower bound of the Hausdorff dimension of repellers given by Cao, Pesin and Zhao in~\cite{caopesinzhao2019}. Besides, we prove that the Carath$\acute{\text{e}}$odory singular dimension of non-dense orbit set equals that of the repeller.