Uniform approximation problems of expanding Markov maps

Abstract

Let $T:[0,1]\to [0,1]$ be an expanding Markov map with a finite partition. Let ${\mu }_{\varphi }$ be the invariant Gibbs measure associated with a Hölder continuous potential $\varphi$. In this paper, we investigate the size of the uniform approximation set \[\mathcal U^κ(x):=\{y\in[0,1]:\forall N\gg1,~\exists n\le N, \text{ such that }|T^nx-y|<N^{-κ}\},\] where $\kappa >0$ and $x\in [0,1]$. The critical value of $\kappa$ such that ${\text{dim}}_{\text{H}}{\mathcal{U}}^{\kappa }(x)=1$ for ${\mu }_{\varphi }$-a.e.$x$ is proven to be $1/{\alpha }_{\max }$, where ${\alpha }_{\max }=-\int \varphi d{\mu }_{\max }/\int \log |T^{\prime }|d{\mu }_{\max }$ and ${\mu }_{\max }$ is the Gibbs measure associated with the potential $-\log |T^{\prime }|$. Moreover, when $\kappa >1/{\alpha }_{\max }$, we show that for ${\mu }_{\varphi }$-a.e.$x$, the Hausdorff dimension of ${\mathcal{U}}^{\kappa }(x)$ agrees with the multifractal spectrum of ${\mu }_{\varphi }$.