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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0056-0,
author = {Piotr Szuca},
title = {F-limit points in dynamical systems defined on the interval},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {170-176},
zbl = {06132922},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0056-0}
}
Piotr Szuca. F-limit points in dynamical systems defined on the interval. Open Mathematics, Tome 11 (2013) pp. 170-176. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0056-0/
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