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Diophantine approximation of the orbits in topological dynamical systems

  • * Corresponding author: Jun Wu

    * Corresponding author: Jun Wu

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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  • 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.

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


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