$\mathscr{F}-$Mean Proximality for Amenable Semigroups Actions

Document Type : Original Article

Authors

Mohammad Akbari Tootkaboni

Javad Jafari

Abbas Sahleh

Department of Pure Mathematics Faculty of Mathematical Sciences University of Guilan Rasht, Iran.

10.22067/kjm.2026.520291.3534

Abstract

In this paper, we introduce the notions of proximality and regionally proximal relations via a Furstenberg family with respect to the semigroup \( S \). We also explore the relationship between \(\mathscr{F}\)-equicontinuity and \(\tau\mathscr{F}\)-equicontinuity. Furthermore, we show that if \(\left(X, \{ T_s \}_{s \in S} \right)\) is \(\mathscr{F}\)-mean equicontinuous, then \( P_{\mathscr{F}}\left(X, \{ T_s \}_{s \in S} \right) \) and \( Q_{\mathscr{F}}\left(X, \{ T_s \}_{s \in S} \right) \) coincide for amenable semigroup actions. We prove that if a countable, discrete, infinite, commutative semigroup action is \(\mathscr{F}\)-mean sensitive, and there exists an \(\mathscr{F}\)-mean proximal pair consisting of a transitive point and a periodic point, then the system is \(\mathscr{F}\)-mean Li--Yorke chaotic.

Keywords

$\mathscr{F}$- mean sensitivity

$\mathscr{F}-$ mean equicontinuity

$\mathscr{F}-$mean proximal

$\mathscr{F}-$regionally proximal

Furstenberg family

Subjects

• 37 Dynamical systems and ergodic theory

---