2016/11/24 by S. Garcı́a-Ferreira, Garcia-Ferreira, S., Y. Rodríguez-López +3 · 1 citation
Mathematics · #54D80 #54G20 #54H20 #Advanced Topology and Set Theory #FOS: Mathematics #Functional Equations Stability Results #General Topology (math.GN) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1611.08290
openalex publication_date 2016/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let E(X,f) be the Ellis semigroup of a dynamical system (X,f) where X is a compact metric space. We analyze the cardinality of E(X,f) for a compact countable metric space X. A characterization when E(X,f) and E(X,f)^* = E(X,f) ∖ \ fn : n ∈ ℕ\ are both finite is given. We show that if the collection of all periods of the periodic points of (X,f) is infinite, then E(X,f) has size 2ℵ0. It is also proved that if (X,f) has a point with a dense orbit and all elements of E(X,f) are continuous, then |E(X,f)| ≤ |X|. For dynamical systems of the form (ω2 +1,f), we show that if there is a point with a dense orbit, then all elements of E(ω2+1,f) are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where E(ω2+1,f) and ω2+1 are homeomorphic but not algebraically homeomorphic, where ω2+1 is taken with the usual ordinal addition as semigroup operation.