×

A survey of some aspects of dynamical topology: dynamical compactness and Slovak spaces. (English) Zbl 1443.37006

This survey is concerned with a field called dynamical topology, which studies topological properties of spaces and maps characterized in topological terms. Three concepts in particular are discussed here. The first is topological properties of the set of all transitive maps of a compact interval. The second is dynamical compactness, which is defined in terms of a time-changed notion of omega-limit set in terms of a Furstenberg family of subsets of the natural numbers. The third is that of a Slovak space, meaning a non-empty compact metric space with the property that its self-homeomorphism group is generated by a single minimal homeomorphism. It is not obvious that such spaces exist, but earlier work of T. Downarowicz et al. [J. Dyn. Differ. Equations 29, No. 1, 243–257 (2017; Zbl 1384.37015)] shows that there are Slovak metric continua, and that there are examples for which the topological entropy of the homeomorphism generating the auto-homeomorphism group has any value in \([0,\infty]\).
Sadly the author of this survey passed away on May 16, 2018. Lubomír Snoha completed the editing of this paper, and the proceedings of the 2018 activities at the Max Planck Institute in Bonn [P. Moree (ed.) et al., Dynamics: topology and numbers. Conference, Max Planck Institute for Mathematics, Bonn, Germany, July 2–6, 2018. Providence, RI: American Mathematical Society (AMS) (2020; Zbl 1448.37001)] were dedicated to Sergiĭ’s memory.

MSC:

37B02 Dynamics in general topological spaces
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B40 Topological entropy
37B45 Continua theory in dynamics
54C05 Continuous maps
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D30 Compactness
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Akin; E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84, 243-286 (2001) · Zbl 1182.37009
[2] E. Akin; E. Glasner, Residual properties and almost equicontinuity, J. Anal. Math., 84, 243-286 (2001) · Zbl 1045.37004
[3] E. Akin; S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16, 1421-1433 (2003) · Zbl 1388.37018
[4] E. Akin; J. Rautio, Chain transitive homeomorphisms on a space: All or none, Pacific J. Math., 291, 1-49 (2017) · Zbl 0654.54027
[5] J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988, Notas de Matemática [Mathematical Notes], 122. · Zbl 1125.37005
[6] J. Auslander; S. Kolyada; L’. Snoha, Functional envelope of a dynamical system, Nonlinearity, 20, 2245-2269 (2007) · Zbl 0448.54040
[7] J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J., (2) 32 (1980), 177-188. · Zbl 0158.41503
[8] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math., 60, 241-249 (1967) · Zbl 1307.37005
[9] T. Das; E. Shah; L’. Snoha, (Non-)expansivity in functional envelopes, J. Math. Anal. Appl., 410, 1043-1048 (2014) · Zbl 0605.22001
[10] T. Dobrowolski, Examples of topological groups homeomorphic to \(l_2^f\), Proc. Amer. Math. Soc., 98, 303-311 (1986) · Zbl 0055.10401
[11] Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc., (3) 4 (1954), 168-176. · Zbl 1096.37002
[12] T. Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and Topological Dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, 7-37. · Zbl 1384.37015
[13] T. Downarowicz; L’. Snoha; D. Tywoniuk, Minimal spaces with cyclic group of homeomorphisms, J. Dynam. Differential Equations, 29, 243-257 (2017) · Zbl 1311.57046
[14] F. T. Farrell and A. Gogolev, The space of Anosov diffeomorphisms, J. Lond. Math. Soc., (2) 89 (2014), 383-396. · Zbl 0443.58011
[15] A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup., 13, 45-93 (1980) · Zbl 0970.37007
[16] B. R. Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on \({{\rm{T}}^5} \), Bol. Soc. Brasil. Mat. (N.S.), 31, 277-285 (2000) · Zbl 0146.28502
[17] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1, 1-49 (1967) · Zbl 0425.54023
[18] P. Gartside; A. Glyn, Autohomeomorphism groups, Topology Appl., 129, 103-110 (2003) · Zbl 1020.54026
[19] H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math., 34 (1978), 61-85 (1979). · Zbl 0790.58025
[20] P. Gartside; A. Glyn, Autohomeomorphism groups, Topology Appl., 129, 103-110 (2003) · Zbl 0084.38402
[21] E. Glasner; B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6, 1067-1075 (1993) · Zbl 0087.37802
[22] J. de Groot; R. J. Wille, Rigid continua and topological group-pictures, Arch. Math., 9, 441-446 (1958) · Zbl 0429.58012
[23] J. de Groot, Groups represented by homeomorphism groups, Math. Ann., 138, 80-102 (1959) · Zbl 0044.12503
[24] J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70, 133-160 (1979) · Zbl 1247.22007
[25] S. Harada, Remarks on the topological group of measure preserving transformations, Proc. Japan Acad., 27, 523-526 (1951)
[26] K. H. Hofmann; S. A. Morris, Compact homeomorphism groups are profinite, Topology Appl., 159, 2453-2462 (2012) · Zbl 1354.37021
[27] K. H. Hofmann and S. A. Morris, Representing a profinite group as the homeomorphism group of a continuum, preprint, arXiv: 1108.3876. · Zbl 1401.37018
[28] W. Huang; D. Khilko; S. Kolyada; G. Zhang, Dynamical compactness and sensitivity, J. Differential Equations, 260, 6800-6827 (2016) · Zbl 1390.37016
[29] W. Huang; D. Khilko; S. Kolyada; A. Peris; G. Zhang, Finite intersection property and dynamical compactness, J. Dynam. Differential Equations, 30, 1221-1245 (2018) · Zbl 0201.56701
[30] W. Huang; S. Kolyada; G. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory Dynam. Systems, 38, 651-665 (2018) · Zbl 1352.37115
[31] M. Keane, Contractibility of the automorphism group of a nonatomic measure space, Proc. Amer. Math. Soc., 26, 420-422 (1970) · Zbl 1376.37082
[32] S. Kolyada; M. Misiurewicz; L’. Snoha, Spaces of transitive interval maps, Ergodic Theory Dynam. Systems, 35, 2151-2170 (2015) · Zbl 1283.37012
[33] S. Kolyada, M. Misiurewicz and L’. Snoha, Loops of transitive interval maps, Dynamics and numbers, Contemp. Math., Amer. Math. Soc., Providence, RI, 669 (2016), 137-154. · Zbl 1316.37013
[34] S. Kolyada; O. Rybak, On the Lyapunov numbers, Colloq. Math., 131, 209-218 (2013) · Zbl 0909.54012
[35] S. Kolyada; J. Semikina, On topological entropy: When positivity implies +infinity, Proc. Amer. Math. Soc., 143, 1545-1558 (2015) · Zbl 0907.54036
[36] S. Kolyada; L’. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4, 205-233 (1996) · Zbl 1031.37014
[37] S. Kolyada; L’. Snoha, Some aspects of topological transitivity - a survey, Iteration Theory (ECIT 94) (Opava), Grazer Math. Ber., 334, 3-35 (1997) · Zbl 1234.37034
[38] S. Kolyada; L’. Snoha; S. Trofimchuk, Noninvertible minimal maps, Fund. Math., 168, 141-163 (2001) · Zbl 1335.54039
[39] J. Li, Transitive points via Furstenberg family, Topology Appl., 158, 2221-2231 (2011) · Zbl 1277.37033
[40] J. Li; X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32, 83-114 (2016) · Zbl 1132.54023
[41] M. Matviichuk, On the dynamics of subcontinua of a tree, J. Difference Equ. Appl., 19, 223-233 (2013) · Zbl 0722.58008
[42] T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20, 2115-2126 (2007) · Zbl 0188.55503
[43] N. T. Nhu, The group of measure preserving transformations of the unit interval is an absolute retract, Proc. Amer. Math. Soc., 110, 515-522 (1990) · Zbl 0927.37004
[44] K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24, 278-280 (1970)
[45] P. Raith, Topological transitivity for expanding piecewise monotonic maps on the interval, Aequationes Math., 57, 303-311 (1999) · Zbl 0174.54402
[46] D. Ruelle, Dynamical systems with turbulent behavior, Mathematical Problems in Theoretical Physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), Lecture Notes in Phys., vol. 80, Springer, Berlin-New York, 1978,341-360.
[47] A. N. Šarkovskiĭ, On attracting and attracted sets, Soviet Math. Dokl., 6, 268-270 (1965)
[48] A. N. Šarkovskiĭ, Continuous mapping on the limit points of an iteration sequence, Ukrain. Mat. Ž., 18, 127-130 (1966) · Zbl 0475.28009
[49] A. N. Šarkovskiĭ, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kiev, 1989. · Zbl 1176.28017
[50] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. · Zbl 0475.28009
[51] T. Yagasaki, Weak extension theorem for measure-preserving homeomorphisms of noncompact manifolds, J. Math. Soc. Japan, 61, 687-721 (2009) · Zbl 1176.28017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.