On sets of pointwise recurrence and dynamically thick sets
Year: 2026
Type: article
Abstract: A set A ⊆ N $A \subseteq \mathbb {N}$ upper A subset of or equal to double struck upper N is a set of pointwise recurrence if for all minimal dynamical systems ( X , T ) $(X, T)$ left parenthesis upper X comma upper T right parenthesis , all x ∈ X $x \in X$ x element of upper X , and all open neighborhoods U ⊆ X $U \subseteq X$ upper U subset of or equal to upper X of x , there exists a time n ∈ A $n \in A$ n element of upper A such that T n x ∈ U $T^n x \in U$ upper T Superscript n Baseline x element of upper U . The set A is dynamically thick if the same holds for all non-empty, open sets U ⊆ X $U \subseteq X$ upper U subset of or equal to upper X . Our main results give combinatorial characterizations of sets of pointwise recurrence and dynamically thick sets that allow us to answer questions of Host, Kra, and Maass [Variations on topological recurrence. Monatsh. Math. 179 (1) (2016), 57–89] and Glasner, Tsankov, Weiss, and Zucker [Bernoulli disjointness. Duke Math. J . 170 (4) (2021), 615–651]. We also introduce and study a local version of dynamical thickness called dynamical piecewise syndeticity . We show that dynamically piecewise syndetic sets are piecewise syndetic, generalizing results of Glasner and Weiss [Interpolation sets for subalgebras of
Authors: DANIEL GLASSCOCK, ANH N. LE
Institutions: University of Massachusetts Lowell, University of Denver
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7171865501