Strong universality, recurrence, and analytic P-ideals in dynamical systems
Paolo Leonetti
TL;DR
This work develops a nonlinear dynamical framework relative to ideals on the natural numbers, introducing strong notions of universality and recurrence via subsequences constrained by non-small index sets. It builds a structural bridge between the complexity of ideals (e.g., $F_\\sigma$ and analytic $P$-ideals) and dynamical properties expressed through $\\mathsf{I}$-cluster/limit points, $\\mathsf{I}$-universal points, and $\\mathsf{I}$-recurrence. By exploiting upper Furstenberg families and return-sets, the authors derive comeager/dense regularity results, characterize equivalences between universal/recurrence notions in various settings, and apply these to nonlinear and linear systems alike. Notably, they establish conditions under which $\\mathsf{I}$-universality aligns with the cluster-point property in open sets, show dense/comeager sets of universal vectors in many contexts, and prove Ansari-type results for arithmetic ideals, thereby broadening the scope of orbital-structure theorems beyond Fin-based hypercyclicity.
Abstract
Given a dynamical system $(X,T)$ and a family $\mathsf{I}\subseteq \mathcal{P}(ω)$ 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 Fréchet 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 $A\subseteq ω$ 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 $A\subseteq ω$ with nonzero upper asymptotic density.