Uniform approximation problems of expanding Markov maps

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 $\mu_\phi$.