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Let $X$ be a compact metric space and $f$ a continuous map from $X$ to $X$. Is it true, that if $f$ is weakly mixing, then the entering time $$N(U,V) = \{n \in \mathbb{N}\mid f^n(U) \cap V \neq \emptyset\}$$ is thick i.e. contains arbitrary large sequences of consecutive integers?

I guess it is true, but I could not find where I read it, if some could send me link to the proof, I would very much appreciate it.

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    $\begingroup$ I assume $U,V$ denote nonempty open subsets of $X$. $\endgroup$
    – YCor
    Commented Oct 15, 2023 at 21:50

1 Answer 1

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By weak mixing you mean that product dynamical system is topologically mixing, right? Then the answer is yes, and in fact the two facts are equivalent. See for instance Proposition 2.4, point (3) of this 2015 paper.

For the proof, the authors refer to Theorem 1.11 in: E. Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, 2003.

(Of course I'm assuming that YCor is right in his comment and that $U,V$ are nonempty open sets.)

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