Uniform approximation problems of expanding Markov maps

TL;DR

This paper studies the size and Hausdorff dimension of uniform approximation sets for expanding Markov maps, revealing a critical exponent related to the multifractal spectrum of the invariant measure.

Contribution

It establishes the critical value of approximation exponent for full Hausdorff dimension and links the dimension of approximation sets to the multifractal spectrum.

Findings

Critical exponent for full Hausdorff dimension is 1/α_max.

Hausdorff dimension of approximation sets matches the multifractal spectrum for κ > 1/α_max.

Dimension results hold for μ_φ-almost every x.

Abstract

Let $ T:[0,1]\to[0,1] $ be an expanding Markov map with a finite partition. Let $ \mu_\phi $ be the invariant Gibbs measure associated with a H\"older continuous potential $ \phi $. In this paper, we investigate the size of the uniform approximation set \[\mathcal U^\kappa(x):=\{y\in[0,1]:\forall N\gg1,~\exists n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\},\] where $ \kappa>0 $ and $ x\in[0,1] $. The critical value of $ \kappa $ such that $ \textrm{dim}_{\textrm H}\mathcal U^\kappa(x)=1 $ for $ \mu_\phi $-a.e.$ \, x $ is proven to be $ 1/\alpha_{\max} $, where $ \alpha_{\max}=-\int \phi\,d\mu_{\max}/\int\log|T'|\,d\mu_{\max} $ and $ \mu_{\max} $ is the Gibbs measure associated with the potential $ -\log|T'| $. Moreover, when $ \kappa>1/\alpha_{\max} $, we show that for $ \mu_\phi $-a.e.$ \, x $, the Hausdorff dimension of $ \mathcal U^\kappa(x) $ agrees with the multifractal spectrum of $ \mu_\phi $.