American Institute of Mathematical Sciences

May  2019, 39(5): 2455-2471. doi: 10.3934/dcds.2019104

Diophantine approximation of the orbits in topological dynamical systems

 1 Faculty of Information Technology, Macau University of Science and Technology, Macau, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China

* Corresponding author: Jun Wu

Received  January 2018 Revised  October 2018 Published  January 2019

Fund Project: This work is supported by the Science and Technology Development Fund of Macau (No. 044/2015/A2 and 0024/2018/A1) and National Nature Science Foundation of China (No. 11471130, 11722105, 11831007)

We would like to present a general principle for the shrinking target problem in a topological dynamical system. More precisely, let
 $(X, d)$
be a compact metric space and
 $T:X\to X$
a continuous transformation on
 $X$
. For any integer valued sequence
 $\{a_n\}$
and
 $y\in X$
, define
 $E_y(\{a_n\}) = \bigcap\limits_{\delta>0}\Big\{x\in X: T^nx\in B_{a_n}(y, \delta), \ {\text{for infinitely often}}\ n\in \mathbb N\Big\},$
the set of points whose orbit can well approximate a given point infinitely often, where
 $B_n(x, r)$
denotes the Bowen-ball. It is shown that
 $h_{\text {top}}(E_y(\{a_n\}), T) = \frac{1}{1+a}h_{\text {top}}(X, T), \ \ {\text{with}}\ a = \liminf\limits_{n\to\infty}\frac{a_n}{n},$
if the system
 $(X, T)$
has the specification property. Here
 $h_{\text {top}}$
denotes the topological entropy. An example is also given to indicate that the specification property required in the above result cannot be weakened even to almost specification.
Citation: Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104
References:

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