Diophantine approximation in dynamical systems concerns the Diophantine properties of the orbits. In classic Diophantine approximation, the powerful mass transference principle established by Beresnevich and Velani provides a general principle to the dimension for a limsup set. In this paper, we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system. More precisely, let (X, T) be a topological dynamical system where X is a compact metric space and $$T:X\rightarrow X$$ is an expanding continuous transformation. Given $$y_o\in X$$ , we consider the following limsup set $$\mathcal {W}(T,f)$$ , driven by the dynamical system (X, T), $$\begin{aligned} \Big \{x\in X: x\in B(z, e^{-S_n(f+\log |T'|)(z)})\ \text {for some}\ z\in T^{-n}y_o\ {\text {with infinitely many}}\ n\in \mathbb {N}\Big \}, \end{aligned}$$ where $$\log |T'|$$ is a function reflecting the local conformality of the transformation T, f is a non-negative continuous function over X, and $$S_n (f+ \log |T'|) (z)$$ denotes the ergodic sum $$(f+ \log |T'|) (z)+\cdots +(f+ \log |T'|) (T^{n-1} z)$$ . By proposing a dynamical ubiquity property assumed on the system (X, T), we obtain that the dimensions of X and $$\mathcal {W}(T,f)$$ are both related to the Bowen-Manning-McCluskey formulae, namely the solution to the pressure functions $$\begin{aligned} \texttt {P}(-t\log |T'|)=0\ \text {and}\ \texttt {P}(-t(\log |T'|+f))=0, \text {respectively}. \end{aligned}$$ We call this phenomenon a dynamical dimension transference principle, because of its partial analogy with the mass transference principle. This general principle unifies and extends some known results which were considered only separatedly before. These include the b-adic expansions, expanding rational maps over Julia sets, inhomogeneous. Diophantine approximation on the triadic Cantor set and finite conformal iterated function systems.

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动态丢番图逼近的动态维数传递原理

动力系统中的丢番图近似涉及轨道的丢番图特性。在经典的丢番图近似中，Beresnevich 和 Velani 建立的强大的传质原理为 limsup 集的维数提供了一般原理。在本文中，我们旨在为一般扩展动力系统中出现的 limsup 集的维数找到一个一般原理。更准确地说，让 (X, T) 是一个拓扑动力系统，其中 X 是一个紧凑的度量空间，而 $$T:X\rightarrow X$$ 是一个扩展的连续变换。给定 $$y_o\in X$$ ，我们考虑以下 limsup 集合 $$\mathcal {W}(T,f)$$ ，由动力系统 (X, T) 驱动， $$\begin{aligned} \大 \{x\in X: x\in B(z, e^{-S_n(f+\log |T' |)(z)})\ \text {for some}\ z\in T^{-n}y_o\ {\text {无限多}}\ n\in \mathbb {N}\Big \}, \ end{aligned}$$ 其中 $$\log |T'|$$ 是反映变换 T 局部共形性的函数，f 是 X 上的非负连续函数，$$S_n (f+ \log |T '|) (z)$$ 表示遍历和 $$(f+ \log |T'|) (z)+\cdots +(f+ \log |T'|) (T^{n-1} z)$美元。通过提出假设在系统 (X, T) 上的动态泛在属性，我们得到 X 和 $$\mathcal {W}(T,f)$$ 的维度都与 Bowen-Manning-McCluskey 公式有关，即压力函数的解 $$\begin{aligned} \texttt {P}(-t\log |T'|)=0\ \text {and}\ \texttt {P}(-t(\log | T'|+f))=0, \text {分别}。\end{aligned}$$ 我们称这种现象为动态维度转移原理，因为它部分类似于传质原理。这个一般原则统一和扩展了一些以前仅分开考虑的已知结果。这些包括 b-adic 扩展，在 Julia 集合上扩展有理映射，非齐次。三元康托集和有限保形迭代函数系统上的丢番图近似。