F-limit points in dynamical systems defined on the interval

• Autorzy: Piotr Szuca
• Języki publikacji: EN

Abstrakty

EN

Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider a question about continuity of the multivalued map x → ω fF(x). We point out some connections between the Baire class of f p and tame dynamical systems, and give some open problems.

Słowa kluczowe

EN

Ideal convergence; Filter convergence; Interval maps; ω-limit sets; Hausdorff metric; Ellis semigroup; Enveloping semigroup; Tame dynamical system

Kategorie tematyczne

• 26A21: Classification of real functions; Baire classification of sets and functions
• 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
• 03E15: Descriptive set theory
• 40A30: Convergence and divergence of series and sequences of functions
• 26A03: Foundations: limits and generalizations, elementary topology of the line
• 40A35: Ideal and statistical convergence

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 1
• Strony: 170-176

Daty

wydano

2013-01-01

online

2012-10-24

Twórcy

autor

Piotr Szuca

• University of Gdańsk

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-012-0056-0