2026/06/03 by S. García-Ferreira, Y. Z. Rodriguez-López, A. H. Tomita
Mathematics · #Fixed Point Theorems Analysis #Mathematical Dynamics and Fractals #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.1007/s00233-026-10629-3
crossref issued 2026/06/03 · crossref published 2026/06/03 · crossref published-online 2026/06/03 · openalex publication_date 2026/06/03 · crossref created 2026/06/03 · openalex created_date 2026/06/04 · openalex updated_date 2026/07/29 · crossref published-print 2026/08/01 · crossref deposited 2026/08/03 · crossref indexed 2026/08/03
Let E(X, f) be the Ellis semigroup of a discrete discrete dynamical system (X, f) where X is a compact countable metric space. Set P(X,f): = \ |\mathcal Of(x)|: x \text is a periodic point of X \ , for an eventually periodic point x∈ X let lx ∈ \mathbb N be its waiting time, and set L(X,f):= \ l ∈ \mathbb N: ∃ x ∈ X(x \text is eventually periodic and l = lx) \ . We give two necessary and sufficient statements equivalent to the assertion “E(X, f) is a compactification of \mathbb N with the discrete topology”. And we also prove the following assertions: If each x∈ X has a finite orbit and L(X, f) and P(X, f) are finite, then |E(X,f)| ≤ ∏ s∈ P(X,f)\0,… , s-1\ + max L(X,f). Besides, if all the periods are relative prime numbers, then |E(X,f)|= ∏ s∈ P(X,f)\0,… , s-1\+max L(X,f). Two necessary conditions on a discrete dynamical system are given in order that its Ellis semigroup be countable. We also include several examples that exemplify the diversity of dynamical properties in relation to the cardinality of the Ellis semigroup.