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 $ μ_ϕ$ be the invariant Gibbs measure associated with a Hölder continuous potential $ ϕ$. 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| 0 $ and $ x\in[0,1] $. The critical value of $ κ$ such that $ \textrm{dim}_{\textrm H}\mathcal U^κ(x)=1 $ for $ μ_ϕ$-a.e.$ \, x $ is proven to be $ 1/α_{\max} $, where $ α_{\max}=-\int ϕ\,dμ_{\max}/\int\log|T'|\,dμ_{\max} $ and $ μ_{\max} $ is the Gibbs measure associated with the potential $ -\log|T'| $. Moreover, when $ κ>1/α_{\max} $, we show that for $ μ_ϕ$-a.e.$ \, x $, the Hausdorff dimension of $ \mathcal U^κ(x) $ agrees with the multifractal spectrum of $ μ_ϕ$.