On Some Sets of Almost Continuous Functions which Locally Aproximate a Fixed Function

Anna Loranty; Ryszard J. Pawlak

doi:10.1515/tmmp-2016-0009

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On Some Sets of Almost Continuous Functions which Locally Aproximate a Fixed Function

Tatra Mountains Mathematical Publications

Volume 65 (2016): Issue 1 (March 2016)

By: Anna Loranty and Ryszard J. Pawlak

Open Access

|Aug 2016

Abstract

In this paper, we investigate topological properties of the sets of almost continuous functions f: [0,1] → [0,1] which locally approximate a fixed function. This approximation is associated with the certain value of entropy of a function at a fixed point of the primal function.

References

[1] ALSEDÁ, L.-LLIBRE, J.-MISIUREWICZ, M.: Combinatorial Dynamics and Entropy in Dimension One (2nd ed.), in: Adv. Ser. Nonlinear Dynam., Vol. 5, World Sci., Singapore, 2000.
Search in Google Scholar
[2] BLANCHARD, F.: Topological chaos: what this means? J. Difference Equ. Appl. 15 (2009), 23-46.10.1080/10236190802385355
Search in Google Scholar
[3] BLANCHARD, F.-HUANG, W.-SNOHA, L’: Topological size of scrambled set, Colloq. Math. 110 (2008), 293-361.10.4064/cm110-2-3
Search in Google Scholar
[4] BLOCK, L. S.-COPPEL, W. A.: Dynamics in one Dimension. Lecture Notes in Math., Vol. 1513, Springer-Verlag, Berlin, 1992.
Search in Google Scholar
[5] BOWEN, R.: Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414; erratum 181 (1973), 509-510.
Search in Google Scholar
[6] ČIKLOVÁ, M.: Dynamical systems generated by functions with connected G_δ graphs, Real Anal. Exchange 30 (2004/2005), 617-638.10.14321/realanalexch.30.2.0617
Search in Google Scholar
[7] DINABURG, E. I.: The relation between topological entropy and metric entropy, Soviet Math. Doklady 11 (1970), 13-16.
Search in Google Scholar
[8] JASTRZĘBSKI, J. M.-.JĘDRZEJEWSKI, J. M.-NATKANIEC, T.: On some subclass of Darboux functions, Fund. Math. 138 (1991), 165-173.10.4064/fm-138-3-165-173
Search in Google Scholar
[9] KORCZAK-KUBIAK, E.-LORANTY, A.-PAWLAK, R. J.: On oo-entropy points in real analysis, Opuscula Math. 34 (2014), 799-812, http://dx.doi.org/10.7494/OpMath.2014.34.4.79910.7494/OpMath.2014.34.4.799
Search in Google Scholar
[10] KORCZAK-KUBIAK, E.-LORANTY, A.-PAWLAK, R. J.: On local problem, of entropy for functions from, Zahorski classes (to appear).
Search in Google Scholar
[11] KWIETNIAK, D.-OPROCHA, P.: Topological entropy and chaos for maps induced on hyperspaces, Chaos Solitons Fractals 33 (2007), 76-86.10.1016/j.chaos.2005.12.033
Search in Google Scholar
[12] LI, J.-YE, X.: Recent development of chaos theory in topological dynamics, arXiv:1503.06425 [math.DS].
Search in Google Scholar
[13] DE MELO, W.-VAN STRIEN, S.: One-Dimensional Dynamics. Springer-Verlag, Berlin, 1993.10.1007/978-3-642-78043-1
Search in Google Scholar
[14] NATKANIEC, T.: Almost continuity, Real Anal. Exchange 17 (1991/92), 462-520.10.2307/44153745
Search in Google Scholar
[15] NATKANIEC, T.: Almost Continuity. Habilitation Thesis, Wyzsza Szkoła Pedagogiczna w Bydgoszczy, Bydgoszcz, 1992.
Search in Google Scholar
[16] PAWLAK, R. J.: On points of extreme chaos for almost continuous functions, Tatra Mt. Math. Publ. 62 (2014).10.1515/tmmp-2015-0012
Search in Google Scholar
[17] PAWLAK, R. J.-LORANTY, A.-BA̡KOWSKA, A.: On the topological entropy of continuous and almost continuous functions, Topology Appl. 158 (2011), 2022-2033. 10.1016/j.topol.2011.06.049
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.1515/tmmp-2016-0009 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 105 - 118

Submitted on: Nov 11, 2015

Published on: Aug 4, 2016

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

almost continuous functions,

approximation of a function,

topology of uniform convergence,

entropy (at a point)

Related subjects:

Mathematics,

General mathematics

© 2016 Anna Loranty, Ryszard J. Pawlak, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 65 (2016): Issue 1 (March 2016)