Return times and synchronous recurrence

Let $(X,f)$ be a discrete dynamical system and let $\mathcal{F}$ be a hereditary upward set of subsets of $\mathbb{N}$. A point $x$ is $\mathcal{F}$-recurrent, if for any open neighborhood $U$ of $x$, return times of $x$ to $U$ are in $\mathcal{F}$, that is $\{n : f^n(X)\}\in \mathcal{F}$. A point $x$ is $\mathcal{F}$-PR if for any $\mathcal{F}$-recurrent point $y$ in any dynamical system $(X,g)$ the pair $(x,y)$ is recurrent for $(X\times Y, f\times g)$. In this talk we will present recent results and open problems related to the $\mathcal{F}$-PR property.

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