F-limit points in dynamical systems defined on the interval

Open Mathematics (2013)

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

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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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1. [1] Bartoszynski T., Judah H., Set Theory, A K Peters, Wellesley, 1995
2. [2] Blass A., Ultrafilters: where topological dynamics = algebra = combinatorics, Topology Proc., 1993, 18, 33–56 Zbl0856.54042
3. [3] Bourgain J., Fremlin D.H., Talagrand M., Pointwise compact sets of Baire-measurable functions, Amer. J. Math., 1978, 100(4), 845–886 http://dx.doi.org/10.2307/2373913 Zbl0413.54016
4. [4] Bruckner A.M., Ceder J., Chaos in terms of the map x → ω(x; f), Pacific J. Math., 1992, 156(1), 63–96 Zbl0728.58020
5. [5] Fedorenko V.V., Šarkovskii A.N., Smítal J., Characterizations of weakly chaotic maps of the interval, Proc. Amer. Math. Soc., 1990, 110(1), 141–148 http://dx.doi.org/10.1090/S0002-9939-1990-1017846-5 Zbl0728.26008
6. [6] García-Ferreira S., Sanchis M., Ultrafilter-limit points in metric dynamical systems, Comment. Math. Univ. Carolin., 2007, 48(3), 465–485 Zbl1199.54194
7. [7] Glasner E., Enveloping semigroups in topological dynamics, Topology Appl., 2007, 154(11), 2344–2363 http://dx.doi.org/10.1016/j.topol.2007.03.009 Zbl1123.54015
8. [8] Glasner E., Megrelishvili M., Hereditarily non-sensitive dynamical systems and linear representations, Colloq. Math., 2006, 104(2), 223–283 http://dx.doi.org/10.4064/cm104-2-5 Zbl1094.54020
9. [9] Glasner E., Megrelishvili M., New algebras of functions on topological groups arising from G-spaces, Fund. Math., 2008, 201(1), 1–51 http://dx.doi.org/10.4064/fm201-1-1 Zbl1161.37014
10. [10] Nuray F., Ruckle W.H., Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl., 2000, 245(2), 513–527 http://dx.doi.org/10.1006/jmaa.2000.6778 Zbl0955.40001
11. [11] Rosenthal H.P., A characterization of Banach spaces containing l 1, Proc. Nat. Acad. Sci. U.S.A., 1974, 71, 2411–2413 http://dx.doi.org/10.1073/pnas.71.6.2411 Zbl0297.46013
12. [12] Schweizer B., Smítal J., Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 1994, 344(2), 737–754 http://dx.doi.org/10.1090/S0002-9947-1994-1227094-X Zbl0812.58062
13. [13] Todorcevic S., Topics in Topology, Lecture Notes in Math., 1652, Springer, Berlin, 1997 Zbl0953.54001

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