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March  2018, 38(3): 1007-1031. doi: 10.3934/dcds.2018043

What is topological about topological dynamics?

Chris Good 1,, and  Sergio Macías 2,

1. 

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

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D. F., C. P. 04510. México

* Corresponding author

Received  September 2016 Revised  September 2017 Published  December 2017

Fund Project: The first author gratefully acknowledge support from the European Union through the H2020-MSCA-IF-2014 project ShadOmIC (SEP-210195797). The second named author was partially supported by DGAPA, UNAM. This author thanks The University of Birmingham, UK, for the support given during this research.

We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definition stated in terms of a metric in compact metric spaces.

We show for example that in a Tychonoff space, transitivity and dense periodic points imply (uniform) sensitivity to initial conditions. We generalise Bryant's result that a compact Hausdorff space admitting a $c$-expansive homeomorphism in the obvious uniform sense is metrizable. We study versions of shadowing, generalising a number of well-known results to the topological setting, and internal chain transitivity, showing for example that $ω$-limit sets are (uniform) internally chain transitive and weak incompressibility is equivalent to (uniform) internal chain transitivity in compact spaces.

Keywords: Compact Hausdorff space, compactum, c-expansive map, dynamical systems, uniform chain recurrent point, uniform internal chain transitivity, uniform pseudo-orbit, uniform shadowing, uniformity.
Mathematics Subject Classification: Primary: 37B99, 37C50, 54B20, 54H20.
Citation: Chris Good, Sergio Macías. What is topological about topological dynamics?. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1007-1031. doi: 10.3934/dcds.2018043
References:
[1]

E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics (1), Amer. Math. Soc., Providence, RI, 1993.

[2]

E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in Convergence in Ergodic Theory and Probability, (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, 5 (1996), 25-40.

[3]

E. Akin, J. Auslander and A. Nagar, Variations on the concept of topological transitivity, Studia Math., 235 (2016), 225-249.  doi: 10.4064/sm8553-7-2016.

[4]

E. Akin and J. D. Carlson, Conceptions of topological transitivity, Topology Appl., 159 (2012), 2815-2830.  doi: 10.1016/j.topol.2012.04.016.

[5]

E. Akin and J. Rautio, Chain transitive homeomorphisms on a space: All or none, Pacific J. Math., 291 (2017), 1-49.  doi: 10.2140/pjm.2017.291.1.

[6]

N. Aoki, Topological dynamics, in K. Morita and J. Nagata, Topics in General Topology, North Holland, Amsterdam, New York, Oxford, Tokyo, 41 (1989), 625-740.

[7]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North Holland, Amsterdam, London, New York, Tokyo, 1994.

[8]

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.

[9]

M. Awartani and S. Elaydi, An extension of chaotic dynamics to general topological spaces, Panamer. Math. J., 10 (2000), 61-71. 

[10]

J. Banks and S. Brett, A note on equivalent definitions of topological transitivity, Discrete Contin. Dyn. Syst., 33 (2013), 1293-1296. 

[11]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stace, On Devaney's definition of chaos, Amer. Math. Month., 99 (1992), 332-334.  doi: 10.2307/2324899.

[12]

A. D. Barwell, ω-Limit Sets of Discrete Dynamical Systems, Ph. D. Dissertation, The University of Birmingham, 2010.

[13]

A. D. Barwell, C. Good and P. Oprocha, Shadowing and expansivity in subspaces, Fund. Math., 219 (2012), 223-243.  doi: 10.4064/fm219-3-2.

[14]

A. D. Barwell, C. Good, P. Oprocha and B. Raines, Characterizations of ω-limit sets in topologically hyperbolic systems, Discrete Contin. Dyn. Syst., 33 (2013), 1819-1833. 

[15]

N. C. Bernardes and U. B. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math., 231 (2012), 1655-1680.  doi: 10.1016/j.aim.2012.05.024.

[16]

E. Bilokopytov and S. F. Kolyada, Transitive maps on topological spaces, Ukrainian Math. J., 65 (2014), 1293-1318.  doi: 10.1007/s11253-014-0860-8.

[17]

W. R. Brian, Abstract omega-limit sets, preprint, Available from: https://wrbrian.files.wordpress.com/2012/01/omegalimitsets.pdf.

[18]

T. A. Brown and W. W. Comfort, New method for expansion and contraction maps in uniform spaces, Proc. Amer. Math. Soc., 11 (1960), 483-486.  doi: 10.1090/S0002-9939-1960-0113210-2.

[19]

B. F. Bryant, On expansive homeomorphisms, Pacific J. Math., 10 (1960), 1163-1167.  doi: 10.2140/pjm.1960.10.1163.

[20]

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.

[21]

M. Edelstein, On nonexpansive mappings of uniform spaces, Indag. Math., 27 (1965), 47-51. 

[22]

R. Engelking, General Topology, Sigma series in pure mathematics, Vol. 6, Heldermann, Berlin, 1989.

[23]

L. Fernández and C. Good, Shadowing for induced maps of hyperspaces, Fund. Math., 235 (2016), 277-286.  doi: 10.4064/fm136-2-2016.

[24]

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

[25]

B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc., 8 (1974), 633-641. 

[26]

W. Huang, S. Kolyada and G. Zhang, Analogues of Auslander-Yorke the-orems for multi-sensitivity, Ergodic Theory Dynamical Systems, published online on 22 September 2016.

[27]

X. Huang, F. Zeng and G. Zhang, Semi-openness and almost-openness of induced mappings, Appl. Math. J. Chinese Univ. Ser. B, 20 (2005), 21-26.  doi: 10.1007/s11766-005-0032-6.

[28]

W. J. Kammerer and R. M. Kasriel, On contractive mappings in uniform spaces, Proc. Amer. Math. Soc., 15 (1964), 288-290.  doi: 10.1090/S0002-9939-1964-0159307-6.

[29]

C. M. Lee, A development of contraction mapping principles on Hausdorff uniform spaces, Trans. Amer. Math. Soc., 226 (1977), 147-159.  doi: 10.1090/S0002-9947-1977-0428315-3.

[30]

E. Marczewski, Séparabilité et multiplication cartésienne des espaces topologiques, Fund. Math., 34 (1947), 127-143.  doi: 10.4064/fm-34-1-127-143.

[31]

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4.

[32]

C. A. Morales and V. Sirvent, Expansivity for measures on uniform spaces, Trans. Amer. Math. Soc., 368 (2016), 5399-5414. 

[33]

W. L. Reddy, Expanding maps on compact metric spaces, Topology Appl., 13 (1982), 327-334.  doi: 10.1016/0166-8641(82)90040-2.

[34]

F. Rhodes, A generalization of isometries to uniform spaces, Proc. Cambridge Philos. Soc., 52 (1956), 399-405.  doi: 10.1017/S0305004100031406.

[35]

S. Ruette, Chaos on the Interval, University Lecture Series, 67. American Mathematical Society, Providence, RI, 2017.

[36]

S. Silverman, On maps with dense orbits and the definition of chaos, Rocky Mountain J. Math., 22 (1992), 353-375.  doi: 10.1216/rmjm/1181072815.

[37]

M. Vellekoop and R. Berglund, On intervals, transitivity =chaos, Amer. Math. Monthly, 101 (1994), 353-355.  doi: 10.2307/2975629.

[38]

T. Wang, J. Yin and Q. Yan, Several transitive properties and Devaney's chaos, Acta Math. Sin. (Engl. Ser.), 32 (2016), 373-383.  doi: 10.1007/s10114-016-5050-1.

[39]

Y. Wang, G. Wei, W. H. Campbell and S. Bourquin, A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology, Chaos Solitons Fractals, 41 (2009), 1708-1717.  doi: 10.1016/j.chaos.2008.07.014.

[40]

S. Willard, General Topology, Addison-Wesley, Reading, Massachusetts; London, 1970.

[41]

K. Yan and F. Zend, Topological entropy, pseudo-orbits and uniform spaces, Topology Appl., 210 (2016), 168-182.  doi: 10.1016/j.topol.2016.07.016.

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics (1), Amer. Math. Soc., Providence, RI, 1993.

[2]

E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in Convergence in Ergodic Theory and Probability, (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, 5 (1996), 25-40.

[3]

E. Akin, J. Auslander and A. Nagar, Variations on the concept of topological transitivity, Studia Math., 235 (2016), 225-249.  doi: 10.4064/sm8553-7-2016.

[4]

E. Akin and J. D. Carlson, Conceptions of topological transitivity, Topology Appl., 159 (2012), 2815-2830.  doi: 10.1016/j.topol.2012.04.016.

[5]

E. Akin and J. Rautio, Chain transitive homeomorphisms on a space: All or none, Pacific J. Math., 291 (2017), 1-49.  doi: 10.2140/pjm.2017.291.1.

[6]

N. Aoki, Topological dynamics, in K. Morita and J. Nagata, Topics in General Topology, North Holland, Amsterdam, New York, Oxford, Tokyo, 41 (1989), 625-740.

[7]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, North Holland, Amsterdam, London, New York, Tokyo, 1994.

[8]

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.

[9]

M. Awartani and S. Elaydi, An extension of chaotic dynamics to general topological spaces, Panamer. Math. J., 10 (2000), 61-71. 

[10]

J. Banks and S. Brett, A note on equivalent definitions of topological transitivity, Discrete Contin. Dyn. Syst., 33 (2013), 1293-1296. 

[11]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stace, On Devaney's definition of chaos, Amer. Math. Month., 99 (1992), 332-334.  doi: 10.2307/2324899.

[12]

A. D. Barwell, ω-Limit Sets of Discrete Dynamical Systems, Ph. D. Dissertation, The University of Birmingham, 2010.

[13]

A. D. Barwell, C. Good and P. Oprocha, Shadowing and expansivity in subspaces, Fund. Math., 219 (2012), 223-243.  doi: 10.4064/fm219-3-2.

[14]

A. D. Barwell, C. Good, P. Oprocha and B. Raines, Characterizations of ω-limit sets in topologically hyperbolic systems, Discrete Contin. Dyn. Syst., 33 (2013), 1819-1833. 

[15]

N. C. Bernardes and U. B. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math., 231 (2012), 1655-1680.  doi: 10.1016/j.aim.2012.05.024.

[16]

E. Bilokopytov and S. F. Kolyada, Transitive maps on topological spaces, Ukrainian Math. J., 65 (2014), 1293-1318.  doi: 10.1007/s11253-014-0860-8.

[17]

W. R. Brian, Abstract omega-limit sets, preprint, Available from: https://wrbrian.files.wordpress.com/2012/01/omegalimitsets.pdf.

[18]

T. A. Brown and W. W. Comfort, New method for expansion and contraction maps in uniform spaces, Proc. Amer. Math. Soc., 11 (1960), 483-486.  doi: 10.1090/S0002-9939-1960-0113210-2.

[19]

B. F. Bryant, On expansive homeomorphisms, Pacific J. Math., 10 (1960), 1163-1167.  doi: 10.2140/pjm.1960.10.1163.

[20]

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.

[21]

M. Edelstein, On nonexpansive mappings of uniform spaces, Indag. Math., 27 (1965), 47-51. 

[22]

R. Engelking, General Topology, Sigma series in pure mathematics, Vol. 6, Heldermann, Berlin, 1989.

[23]

L. Fernández and C. Good, Shadowing for induced maps of hyperspaces, Fund. Math., 235 (2016), 277-286.  doi: 10.4064/fm136-2-2016.

[24]

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

[25]

B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc., 8 (1974), 633-641. 

[26]

W. Huang, S. Kolyada and G. Zhang, Analogues of Auslander-Yorke the-orems for multi-sensitivity, Ergodic Theory Dynamical Systems, published online on 22 September 2016.

[27]

X. Huang, F. Zeng and G. Zhang, Semi-openness and almost-openness of induced mappings, Appl. Math. J. Chinese Univ. Ser. B, 20 (2005), 21-26.  doi: 10.1007/s11766-005-0032-6.

[28]

W. J. Kammerer and R. M. Kasriel, On contractive mappings in uniform spaces, Proc. Amer. Math. Soc., 15 (1964), 288-290.  doi: 10.1090/S0002-9939-1964-0159307-6.

[29]

C. M. Lee, A development of contraction mapping principles on Hausdorff uniform spaces, Trans. Amer. Math. Soc., 226 (1977), 147-159.  doi: 10.1090/S0002-9947-1977-0428315-3.

[30]

E. Marczewski, Séparabilité et multiplication cartésienne des espaces topologiques, Fund. Math., 34 (1947), 127-143.  doi: 10.4064/fm-34-1-127-143.

[31]

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152-182.  doi: 10.1090/S0002-9947-1951-0042109-4.

[32]

C. A. Morales and V. Sirvent, Expansivity for measures on uniform spaces, Trans. Amer. Math. Soc., 368 (2016), 5399-5414. 

[33]

W. L. Reddy, Expanding maps on compact metric spaces, Topology Appl., 13 (1982), 327-334.  doi: 10.1016/0166-8641(82)90040-2.

[34]

F. Rhodes, A generalization of isometries to uniform spaces, Proc. Cambridge Philos. Soc., 52 (1956), 399-405.  doi: 10.1017/S0305004100031406.

[35]

S. Ruette, Chaos on the Interval, University Lecture Series, 67. American Mathematical Society, Providence, RI, 2017.

[36]

S. Silverman, On maps with dense orbits and the definition of chaos, Rocky Mountain J. Math., 22 (1992), 353-375.  doi: 10.1216/rmjm/1181072815.

[37]

M. Vellekoop and R. Berglund, On intervals, transitivity =chaos, Amer. Math. Monthly, 101 (1994), 353-355.  doi: 10.2307/2975629.

[38]

T. Wang, J. Yin and Q. Yan, Several transitive properties and Devaney's chaos, Acta Math. Sin. (Engl. Ser.), 32 (2016), 373-383.  doi: 10.1007/s10114-016-5050-1.

[39]

Y. Wang, G. Wei, W. H. Campbell and S. Bourquin, A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology, Chaos Solitons Fractals, 41 (2009), 1708-1717.  doi: 10.1016/j.chaos.2008.07.014.

[40]

S. Willard, General Topology, Addison-Wesley, Reading, Massachusetts; London, 1970.

[41]

K. Yan and F. Zend, Topological entropy, pseudo-orbits and uniform spaces, Topology Appl., 210 (2016), 168-182.  doi: 10.1016/j.topol.2016.07.016.

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