April  2020, 40(4): 2441-2474. doi: 10.3934/dcds.2020121

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

Received  July 2019 Revised  November 2019 Published  January 2020

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.

Citation: Chris Good, Robert Leek, Joel Mitchell. Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2441-2474. doi: 10.3934/dcds.2020121
References:
[1]

E. AkinJ. 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.   Google Scholar

[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.  Google Scholar

[3]

J. AuslanderG. Greschonig and A. Nagar, Reflections on equicontinuity, Proc. Amer. Math. Soc., 142 (2014), 3129-3137.  doi: 10.1090/S0002-9939-2014-12034-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. CorbachoV. 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.  Google Scholar

[9]

E. CorbachoV. Tarieladze and R. Vidal, Observations about equicontinuity and related concepts, Topology Appl., 156 (2009), 3062-3069.  doi: 10.1016/j.topol.2009.02.011.  Google Scholar

[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.  Google Scholar

[11] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity. Westview Press, Boulder, CO, 2003.   Google Scholar
[12]

R. Engelking, General Topology, Second edition, Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989.  Google Scholar

[13]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067–1075. doi: 10.1088/0951-7715/6/6/014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

W. HuangS. 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.  Google Scholar

[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.  Google Scholar

[19]

J. L. Kelley, General Topology, Graduate Texts in Mathematics, No. 27. Springer-Verlag, New York-Berlin, 1975.  Google Scholar

[20]

J. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Mitchell, Orbital shadowing, $\omega$-limit sets and minimality, Topology Appl., 268 (2019), 106903, 7 pp. doi: 10.1016/j.topol.2019.106903.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. L. Royden, Real Analysis, Third edition. Macmillan Publishing Company, New York, 1988.  Google Scholar

[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.  Google Scholar

[30]

T. WangJ. 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.  Google Scholar

[31]

S. Willard, General Topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970.  Google Scholar

[32]

X. X. WuY. LuoX. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

E. AkinJ. 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.   Google Scholar

[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.  Google Scholar

[3]

J. AuslanderG. Greschonig and A. Nagar, Reflections on equicontinuity, Proc. Amer. Math. Soc., 142 (2014), 3129-3137.  doi: 10.1090/S0002-9939-2014-12034-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. CorbachoV. 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.  Google Scholar

[9]

E. CorbachoV. Tarieladze and R. Vidal, Observations about equicontinuity and related concepts, Topology Appl., 156 (2009), 3062-3069.  doi: 10.1016/j.topol.2009.02.011.  Google Scholar

[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.  Google Scholar

[11] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Studies in Nonlinearity. Westview Press, Boulder, CO, 2003.   Google Scholar
[12]

R. Engelking, General Topology, Second edition, Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989.  Google Scholar

[13]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067–1075. doi: 10.1088/0951-7715/6/6/014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

W. HuangS. 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.  Google Scholar

[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.  Google Scholar

[19]

J. L. Kelley, General Topology, Graduate Texts in Mathematics, No. 27. Springer-Verlag, New York-Berlin, 1975.  Google Scholar

[20]

J. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. Mitchell, Orbital shadowing, $\omega$-limit sets and minimality, Topology Appl., 268 (2019), 106903, 7 pp. doi: 10.1016/j.topol.2019.106903.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

H. L. Royden, Real Analysis, Third edition. Macmillan Publishing Company, New York, 1988.  Google Scholar

[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.  Google Scholar

[30]

T. WangJ. 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.  Google Scholar

[31]

S. Willard, General Topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970.  Google Scholar

[32]

X. X. WuY. LuoX. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  A non-sensitive, eventually sensitive system
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