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December 2020 On the density of periodic measures for piecewise monotonic maps and their coding spaces
Kenichiro Yamamoto
Tsukuba J. Math. 44(2): 309-324 (December 2020). DOI: 10.21099/tkbjm/20204402309

Abstract

We prove that for all transitive piecewise monotonic maps, the following conditions are equivalent:

(1) All ergodic measures on $[0,1]$ are approximated by periodic ones;

(2) All invariant measures on coding spaces are approximated by periodic ones;

(3) All ergodic measures on coding spaces are approximated by periodic ones.

If we further assume that the map is piecewise increasing and either right or left continuous, then the following condition is also equivalent to (1)–(3).

(4) All invariant measures on $[0,1]$ are approximated by periodic ones.

We also construct an example of a piecewise decreasing and right continuous map which satisfies (1)–(3), but does not satisfy (4).

Citation

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Kenichiro Yamamoto. "On the density of periodic measures for piecewise monotonic maps and their coding spaces." Tsukuba J. Math. 44 (2) 309 - 324, December 2020. https://doi.org/10.21099/tkbjm/20204402309

Information

Published: December 2020
First available in Project Euclid: 12 April 2021

Digital Object Identifier: 10.21099/tkbjm/20204402309

Subjects:
Primary: 37B10 , 37E05

Keywords: coding space , periodic measure , piecewise monotonic map

Rights: Copyright © 2020 University of Tsukuba, Institute of Mathematics

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Vol.44 • No. 2 • December 2020
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