Diophantine approximation by orbits of Markov maps

TL;DR

This paper explores the Hausdorff dimension of sets of points in dynamical systems, specifically expanding Markov maps, that are well-approximated by their orbits, extending shrinking targets theory using multifractal analysis.

Contribution

It introduces a new approach to quantify approximation sets in Markov maps via multifractal properties of Gibbs measures, expanding the scope of shrinking targets theory.

Findings

Dimensions are characterized using multifractal spectra.

Results extend shrinking targets theory to Markov maps.

Provides explicit formulas for Hausdorff dimensions.

Abstract

In 1995, Hill and Velani introduced the shrinking targets theory. Given a dynamical system $([0,1],T)$, they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well-approximated by orbits $\{T^n x\}_{n\geq 0}$, where $T$ is an expanding Markov map with a finite partition supported by $[0,1]$. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.