F-limit points in dynamical systems defined on the interval

Piotr Szuca

Open Mathematics (2013)

  • Volume: 11, Issue: 1, page 170-176
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Piotr Szuca. "F-limit points in dynamical systems defined on the interval." Open Mathematics 11.1 (2013): 170-176. <http://eudml.org/doc/269273>.

@article{PiotrSzuca2013,
abstract = {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.},
author = {Piotr Szuca},
journal = {Open Mathematics},
keywords = {Ideal convergence; Filter convergence; Interval maps; ω-limit sets; Hausdorff metric; Ellis semigroup; Enveloping semigroup; Tame dynamical system; ideal convergence; filter convergence; interval maps; $\omega $-limit sets; enveloping semigroup; tame dynamical system},
language = {eng},
number = {1},
pages = {170-176},
title = {F-limit points in dynamical systems defined on the interval},
url = {http://eudml.org/doc/269273},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Piotr Szuca
TI - F-limit points in dynamical systems defined on the interval
JO - Open Mathematics
PY - 2013
VL - 11
IS - 1
SP - 170
EP - 176
AB - 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.
LA - eng
KW - Ideal convergence; Filter convergence; Interval maps; ω-limit sets; Hausdorff metric; Ellis semigroup; Enveloping semigroup; Tame dynamical system; ideal convergence; filter convergence; interval maps; $\omega $-limit sets; enveloping semigroup; tame dynamical system
UR - http://eudml.org/doc/269273
ER -

References

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