A dynamical dimension transference principle for dynamical diophantine approximation

Abstract

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.