A note on uniform entropy for maps having topological specification property

Sejal Shah, Ruchi Das, Tarun Das

Abstract

We prove that if a uniformly continuous self-map $f$ of a uniform space has topological specification property then the map $f$ has positive uniform entropy, which extends the similar known result for homeomorphisms on compact metric spaces having specification property. An example is also provided to justify that the converse is not true.


Keywords

topological specification property; uniform entropy; uniform spaces.

Subject classification

37B40; 37B20.

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References

R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.

(http://dx.doi.org/10.1090/S0002-9947-1965-0175106-9)

J. Amigó, K. Keller and V. Unakafova, On entropy, entropy-like quantities, and applications, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), 3301-3343.

(http://dx.doi.org/10.3934/dcdsb.2015.20.3301)

N. Aoki, Topological dynamics, Topics in general topology. Amsterdam, North-Holland Publishing Co. 41 (1989), 625-740.

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414.

(http://dx.doi.org/10.1090/S0002-9947-1971-0274707-X)

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971) 377-397.

T. Ceccherini-Silberstein and M. Coornaert, Sensitivity and Devaney's chaos in uniform spaces, J. Dyn. Control Syst. 19 (2013), 349-357.

(http://dx.doi.org/10.1007/s10883-013-9182-7)

T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces, Topology Appl. 160 (2013), 149-158.

(http://dx.doi.org/10.1016/j.topol.2012.10.010)

D. Dikranjan, M. Sanchis and S. Virili, New and old facts about entropy in uniform spaces and topological groups, Topology Appl. 159 (2012), 1916-1942.

(http://dx.doi.org/10.1016/j.topol.2011.05.046)

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1-49.

(http://dx.doi.org/10.1007/BF01692494)

T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971), 176-180.

(http://dx.doi.org/10.1112/blms/3.2.176)

J. Hofer, Topological entropy for noncompact spaces, Michigan Math. J. 21 (1974), 235-242.

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

(http://dx.doi.org/10.1112/jlms/s2-8.4.633)

J. Kelley, General Topology, D. Van Nostrand Company (1955).

T. Kimura, Completion theorem for uniform entropy, Comment. Math. Univ. Carolin. 39 (1998), 389-399.

S. Shah, R. Das and T. Das, Specification property for topological spaces, J. Dyn. Control Syst., to appear.

(http://dx.doi.org/10.1007/s10883-015-9275-6)

B. Weiss, Single Orbit Dynamics, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI 95 (2000).

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