A note on measure and expansiveness on uniform spaces

Pramod Das; Tarun Das

doi:10.4995/agt.2019.8843

Authors

Pramod Das University of Delhi
Tarun Das University of Delhi

DOI:

https://doi.org/10.4995/agt.2019.8843

Keywords:

expansiveness, measure expansiveness, expansive measures, equicontinuity, shadowing, specification

Abstract

We prove that the set of points doubly asymptotic to a point has measure zero with respect to any expansive outer regular measure for a bi-measurable map on a separable uniform space. Consequently, we give a class of measures which cannot be expansive for Denjoy home-omorphisms on S¹. We then investigate the existence of expansive measures for maps with various dynamical notions. We further show that measure expansive (strong measure expansive) homeomorphisms with shadowing have periodic (strong periodic) shadowing. We relate local weak specification and periodic shadowing for strong measure expansive systems.

Downloads

Download data is not yet available.

Author Biographies

Pramod Das, University of Delhi

Department of Mathematics, Faculty of Mathematical Sciences

Tarun Das, University of Delhi

Department of Mathematics, Faculty of Mathematical Sciences

References

A. Arbieto and C. A. Morales, Some properties of positive entropy maps, Ergodic Theory Dynam. Systems 34 (2014), 765-776. https://doi.org/10.1017/etds.2012.162

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

J. R. Brown, Ergodic theory and topological dynamics, Academic Press (1976).

M. Brin and G. Stuck, Introduction to dynamical systems, Cambridge University Press (2002). https://doi.org/10.1017/CBO9780511755316

B. Carvalho and W. Cordeiro, N-expansive homeomorphisms with the shadowing property, J. Differential Equations 261 (2016), 3734-3755. https://doi.org/10.1016/j.jde.2016.06.003

W. Cordiero, M. Denker and X. Zhang, On specification and measure expansiveness, Discrete Continuous Dyn. Syst. 37 (2017), 1941-1957. https://doi.org/10.3934/dcds.2017082

T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homeomorphisms on non-compact and non-metrizable spaces, Topology Appl. 160 (2013), 149-158. https://doi.org/10.1016/j.topol.2012.10.010

P. Das and T. Das, Various types of shadowing and specification on uniform spaces, J. Dyn. Control Syst. 24 (2018), 253-267. https://doi.org/10.1007/s10883-017-9388-1

M. B. Feldman, A proof of Lusin's theorem, Amer. Math. Month. 88 (1981), 191-192. https://doi.org/10.1080/00029890.1981.11995222

J. F. Jacobsen and W. R. Utz, The non-existence of expansive homeomorphisms on a closed 2-cell, Pacific J. Math. 10 (1960), 1319-1321. https://doi.org/10.2140/pjm.1960.10.1319

I. M. James, Uniform and topological spaces, Springer-Verlag (1994).

J. Kelley, General topology, Van Nostrand Company (1955).

J. D. Knowles, On the existence of non-atomic measures, Mathematika 14 (1967), 62-67. https://doi.org/10.1112/S0025579300008020

C. A. Morales, Measure-expansive systems, Preprint, IMPA, D083 (2011).

C. A. Morales and V. Sirvent, Expansivity for measures on uniform spaces, Trans. Amer. Math. Soc. 368 (2016), 5399-5414. https://doi.org/10.1090/tran/6555

K. R. Parthasarathy, R. R. Ranga and S. R. S. Varadhan, On the category of indecomposable distributions on topological groups, Trans. Amer. Math. Soc. 102 (1962), 200-217. https://doi.org/10.1090/S0002-9947-1962-0153041-7

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. https://doi.org/10.1090/S0002-9939-1950-0038022-3

R. Williams, Some theorems on expansive homeomorphisms, Amer. Math. Month. 8 (1966), 854-856. https://doi.org/10.2307/2314180