Strong universality, recurrence, and analytic P-ideals in dynamical systems

Abstract

Given a dynamical system $(X,T)$ and a family $\mathsf{I}\subseteq \mathcal{P}(\omega)$ of "small" sets of nonnegative integers, a point $x \in X$ is said to be $\mathsf{I}$-strong universal if for each $y \in X$ there exists a subsequence $(T^nx: n \in A)$ of its orbit which is convergent to $y$ and, in addition, the set of indexes $A$ is "not small," that is, $A\notin \mathsf{I}$. An analoguous definition is given for $\mathsf{I}$-strong recurrence. In this work, we provide several structural properties and relationships between $\mathsf{I}$-strong universality, $\mathsf{I}$-strong recurrence, and the corresponding ordinary notions of $\mathsf{I}$-universality and $\mathsf{I}$-recurrence. As applications, we provide sufficient conditions which ensure the equivalence between the above notions and the property that each nonempty open set contains some cluster point of some orbit. In addition, we show that if $T$ is a homomorphism on a Fr\'{e}chet space $X$ and there exists a dense set of vectors with null orbit, then for each $y \in X$ the set of all vectors $x \in X$ such that $\lim_{n \in A}T^nx=y$ for some $A\subseteq \omega$ with nonzero upper asymptotic density is either empty or comeager. In the special case of linear dynamical systems on Banach spaces with a dense set of uniformly recurrent vectors, we obtain that $T$ is upper frequently hypercyclic if and only if there exists a hypercyclic vector $x \in X$ for which $\lim_{n \in A}T^nx=0$ for some $A\subseteq \omega$ with nonzero upper asymptotic density.