Dynamic properties of the dynamical system SFnm(X), SFnm(f))

Franco Barragán, Alicia Santiago-Santos, Jesús F. Tenorio

Abstract

Let X be a continuum and let n be a positive integer. We consider the hyperspaces Fn(X) and SFn(X). If m is an integer such that n > m ≥ 1, we consider the quotient space SFnm(X). For a given map f : X → X, we consider the induced maps Fn(f) : Fn(X) → Fn(X), SFn(f) : SFn(X) → SFn(X) and SFnm(f) : SFnm(X) → SFnm(X). In this paper, we introduce the dynamical system (SFnm(X), SFnm (f)) and we investigate some relationships between the dynamical systems (X, f), (Fn(X), Fn(f)), (SFn(X), SFn(f)) and (SFnm(X), SFnm(f)) when these systems are: exact, mixing, weakly mixing, transitive, totally transitive, strongly transitive, chaotic, irreducible, feebly open and turbulent.


Keywords

chaotic; continuum; dynamical system; exact; feebly open; hyperspace; induced map; irreducible; mixing; strongly transitive; symmetric product; symmetric product suspension; totally transitive; transitive; turbulent; weakly mixing

Subject classification

54B20; 37B45; 54F50; 54F15.

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