On topological entropy: When positivity implies +infinity
HTML articles powered by AMS MathViewer
- by Sergiǐ Kolyada and Julia Semikina PDF
- Proc. Amer. Math. Soc. 143 (2015), 1545-1558 Request permission
Abstract:
In this paper we study the relations between the properties of the topological semigroup of all continuous selfmaps $S(X)$ on a compact metric space $X$ (the topological group $H(X)$ of all homeomorphisms on $X$) and possible values of the topological entropy of its elements. In particular, we prove that topological entropy of a functional envelope on the space of all continuous selfmaps on Peano continua or on compact metric spaces with continuum many connected components has only two possible values $0$ and $+\infty$.References
- Joseph Auslander, Sergii Kolyada, and Ľubomír Snoha, Functional envelope of a dynamical system, Nonlinearity 20 (2007), no. 9, 2245–2269. MR 2351032, DOI 10.1088/0951-7715/20/9/012
- L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513, DOI 10.1007/BFb0084762
- H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249. MR 220249, DOI 10.4064/fm-60-3-241-249
- J. de Groot and R. J. Wille, Rigid continua and topological group-pictures, Arch. Math. 9 (1958), 441–446. MR 101514, DOI 10.1007/BF01898628
- J. de Groot, Groups represented by homeomorphism groups, Math. Ann. 138 (1959), 80–102. MR 119193, DOI 10.1007/BF01369667
- Paul Gartside and Aneirin Glyn, Autohomeomorphism groups, Topology Appl. 129 (2003), no. 2, 103–110. MR 1961392, DOI 10.1016/S0166-8641(02)00140-2
- Christian Grillenberger, Constructions of strictly ergodic systems. I. Given entropy, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 323–334. MR 340544, DOI 10.1007/BF00537161
- Karl H. Hofmann and Sidney A. Morris, Compact homeomorphism groups are profinite, Topology Appl. 159 (2012), no. 9, 2453–2462. MR 2921833, DOI 10.1016/j.topol.2011.09.050
- K.H. Hofmann and S. A. Morris, Representing a Profinite Group as the Homeomorphism Group of a Continuum, arXiv:1108.3876v1 [math.GN], 2011.
- John L. Kelley, General topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]. MR 0370454
- Sergiĭ Kolyada, Topologīchna dinamīka: mīnīmal′nīst′, entropīya ta khaos, Pratsī Īnstitutu Matematiki Natsīonal′noï Akademīï Nauk Ukraïni. Matematika ta ïï Zastosuvannya [Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications], vol. 89, Natsīonal′na Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 2011 (Ukrainian, with Ukrainian summary). MR 3089342
- Sergiĭ Kolyada and Ľubomír Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205–233. MR 1402417
- Mykola Matviichuk, Entropy of induced maps for one-dimensional dynamics, Iteration theory (ECIT ’08), Grazer Math. Ber., vol. 354, Institut für Mathematik, Karl-Franzens-Universität Graz, Graz, 2009, pp. 180–185. MR 2649016
- MichałMisiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778
- S. B. Myers, Equicontinuous sets of mappings, Ann. of Math. (2) 47 (1946), 496–502. MR 17526, DOI 10.2307/1969088
- Sam B. Nadler Jr., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR 1192552
- W. Sierpiński, Sur les projections des ensembles complémentaires aux ensembles (A). Fund. Math., 11(1928), 117–122.
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
- Gordon Thomas Whyburn, Analytic topology, American Mathematical Society Colloquium Publications, Vol. XXVIII, American Mathematical Society, Providence, R.I., 1963. MR 0182943
- Koichi Yano, A remark on the topological entropy of homeomorphisms, Invent. Math. 59 (1980), no. 3, 215–220. MR 579700, DOI 10.1007/BF01453235
Additional Information
- Sergiǐ Kolyada
- Affiliation: Institute of Mathematics, NASU, Tereshchenkivs’ka 3, 01601 Kyiv, Ukraine
- MR Author ID: 196872
- Email: skolyada@imath.kiev.ua
- Julia Semikina
- Affiliation: Mathematical Institute of the University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
- Email: julia.semikina@gmail.com
- Received by editor(s): March 9, 2013
- Received by editor(s) in revised form: June 2, 2013
- Published electronically: December 4, 2014
- Additional Notes: The first author was supported by Max-Planck-Institut für Mathematik (Bonn); he acknowledges the hospitality of the Institute
The second author was supported by Bonn International Graduate School in Mathematics - Communicated by: Yingfei Yi
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 1545-1558
- MSC (2010): Primary 37B40; Secondary 54H20, 54H15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12282-9
- MathSciNet review: 3314068