Uniform approximation problems of expanding Markov maps

Abstract

AbstractLet $ 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 Hölder continuous potential $ \phi $ . For $ x\in [0,1] $ and $ \kappa>0 $ , we investigate the size of the uniform approximation set $$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}.\end{align*} $$ The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $ -almost every (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 $ .