Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited

Chris Good; Robert Leek; Joel Mitchell

doi:10.3934/dcds.2020121

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2020, Volume 40, Issue 4: 2441-2474. Doi: 10.3934/dcds.2020121

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Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited

School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

^* Corresponding author: jsm140@bham.ac.uk
^* Corresponding author: jsm140@bham.ac.uk

Received: June 30, 2019

Revised: October 31, 2019

Published: December 2019

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Abstract

We study sensitivity, topological equicontinuity and even continuity in dynamical systems. In doing so we provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. We define what it means for a system to be eventually sensitive; we give a dichotomy for transitive dynamical systems in relation to eventual sensitivity. Along the way we define a property called splitting and discuss its relation to some existing notions of chaos. The approach we take is topological rather than metric.

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Mathematics Subject Classification: Primary: 37B99, 54H20; Secondary: 37B10.

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Figure 1. A non-sensitive, eventually sensitive system

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[1]	E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, 5 (1996), 25-40.
[2]	E. Akin and J. D. Carlson, Conceptions of topological transitivity, Topology Appl., 159 (2012), 2815-2830. doi: 10.1016/j.topol.2012.04.016.
[3]	J. Auslander, G. Greschonig and A. Nagar, Reflections on equicontinuity, Proc. Amer. Math. Soc., 142 (2014), 3129-3137. doi: 10.1090/S0002-9939-2014-12034-X.
[4]	J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2), 32 (1980), 177-188. doi: 10.2748/tmj/1178229634.
[5]	J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332–334. doi: 10.1080/00029890.1992.11995856.
[6]	V. Bergelson, Minimal idempotents and ergodic Ramsey theory, Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 310 (2003), 8-39. doi: 10.1017/CBO9780511546716.004.
[7]	T. Ceccherini-Silberstein and M. Coornaert, Sensitivity and Devaney's chaos in uniform spaces, J. Dyn. Control Syst., 19 (2013), 349-357. doi: 10.1007/s10883-013-9182-7.
[8]	E. Corbacho, V. Tarieladze and R. Vidal, Even continuity and topological equicontinuity in topologized semigroups, Topology Appl., 156 (2009), 1289-1297. doi: 10.1016/j.topol.2008.12.027.
[9]	E. Corbacho, V. Tarieladze and R. Vidal, Observations about equicontinuity and related concepts, Topology Appl., 156 (2009), 3062-3069. doi: 10.1016/j.topol.2009.02.011.
[10]	J. de Vries, Topological Dynamical Systems. An Introduction to the Dynamics of Continuous Mappings, De Gruyter Studies in Mathematics, 59. De Gruyter, Berlin, 2014.
[11]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity. Westview Press, Boulder, CO, 2003.
[12]	R. Engelking, General Topology, Second edition, Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989.
[13]	E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067–1075. doi: 10.1088/0951-7715/6/6/014.
[14]	C. Good and S. Macías, What is topological about topological dynamics?, Discrete Contin. Dyn. Syst., 38 (2018), 1007-1031. doi: 10.3934/dcds.2018043.
[15]	C. Good, J. Mitchell and J. Thomas, Preservation of shadowing in discrete dynamical systems, J. Math. Anal. Appl., 485 (2020), 123767. doi: 10.1016/j.jmaa.2019.123767.
[16]	B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc. (2), 8 (1974), 633-641. doi: 10.1112/jlms/s2-8.4.633.
[17]	W. Huang, S. Kolyada and G. H. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory Dynam. Systems, 38 (2018), 651-665. doi: 10.1017/etds.2016.48.
[18]	W. Huang and X. D. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259-272. doi: 10.1016/S0166-8641(01)00025-6.
[19]	J. L. Kelley, General Topology, Graduate Texts in Mathematics, No. 27. Springer-Verlag, New York-Berlin, 1975.
[20]	J. Li, S. M. Tu and X. D. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory Dynam. Systems, 35 (2015), 2587-2612. doi: 10.1017/etds.2014.41.
[21]	J. Li and X. D. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114. doi: 10.1007/s10114-015-4574-0.
[22]	R. S. Li, A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2815-2823. doi: 10.1016/j.cnsns.2011.11.015.
[23]	R. S. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45 (2012), 753-758. doi: 10.1016/j.chaos.2012.02.003.
[24]	T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985–992. doi: 10.1080/00029890.1975.11994008.
[25]	J. Mitchell, Orbital shadowing, $\omega$-limit sets and minimality, Topology Appl., 268 (2019), 106903, 7 pp. doi: 10.1016/j.topol.2019.106903.
[26]	T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126. doi: 10.1088/0951-7715/20/9/006.
[27]	C. A. Morales and V. Sirvent, Expansivity for measures on uniform spaces, Trans. Amer. Math. Soc., 368 (2016), 5399-5414. doi: 10.1090/tran/6555.
[28]	H. L. Royden, Real Analysis, Third edition. Macmillan Publishing Company, New York, 1988.
[29]	M. Salman and R. Das, Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems, Chaos Solitons Fractals, 115 (2018), 341-348. doi: 10.1016/j.chaos.2018.07.031.
[30]	T. Wang, J. D. Yin and Q. Yan, The sufficient conditions for dynamical systems of semigroup actions to have some stronger forms of sensitivities, J. Nonlinear Sci. Appl., 9 (2016), 989-997. doi: 10.22436/jnsa.009.03.27.
[31]	S. Willard, General Topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970.
[32]	X. X. Wu, Y. Luo, X. Ma and T. X. Lu, Rigidity and sensitivity on uniform spaces, Topology Appl., 252 (2019), 145-157. doi: 10.1016/j.topol.2018.11.014.
[33]	K. S. Yan and F. P. Zeng, Topological entropy, pseudo-orbits and uniform spaces, Topology Appl., 210 (2016), 168-182. doi: 10.1016/j.topol.2016.07.016.