Diophantine approximation of the orbits in topological dynamical systems

Chao Ma; Baowei Wang; Jun Wu

doi:10.3934/dcds.2019104

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2019, Volume 39, Issue 5: 2455-2471. Doi: 10.3934/dcds.2019104

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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
^* Corresponding author: Jun Wu

Received: December 31, 2017

Revised: September 30, 2018

Published: January 2019

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 11J83; Secondary: 28D20.

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