论文标题
扩展马尔可夫地图的统一近似问题
Uniform approximation problems of expanding Markov maps
论文作者
论文摘要
令$ t：[0,1] \至[0,1] $是带有有限分区的扩展马尔可夫地图。令$μ_=$为与Hölder连续潜在$ ϕ $相关的不变吉布斯度量。在本文中，我们研究了均匀近似集的大小 \ [\ [\ Mathcal U^κ（x）：= \ {y \ in [0,1]：\ forall n \ gg1，〜\ arists n \ le n，\ text {sO} 其中$κ> 0 $和$ x \ in [0,1] $。 $κ$的临界价值使得$ \ textrm {dim} _ {\ textrm h} \ Mathcal u^κ（x）= 1 $ for $μ_x$ -a.e. $ -a.e. $ \ $ \，x $已被证明为$ 1/α_ {\ max} $，$ 1/α_ {\ max} $ ϕ \，dμ_ {\ max}/\ int \ log | t'| \，dμ_ {\ max} $和$μ_ {\ max} $是与潜在$ - \ log log | t'|相关的gibbs度量。 $。此外，当$κ> 1/α_ {\ max} $时，我们表明，对于$μ_x$ -A.E. $ \，x $，$ \ MATHCAL u^κ（x）$的Hausdorff尺寸，与$μ__$μ_V的多fractal频谱同意。
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 \[\mathcal U^κ(x):=\{y\in[0,1]:\forall N\gg1,~\exists n\le N, \text{ such that }|T^nx-y|<N^{-κ}\},\] where $ κ>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$.
