| Issue |
Math. Model. Nat. Phenom.
Volume 15, 2020
Systems with Hysteresis and Switching
|
|
|---|---|---|
| Article Number | 15 | |
| Number of page(s) | 14 | |
| DOI | https://doi.org/10.1051/mmnp/2019041 | |
| Published online | 12 March 2020 | |
Invariant measures for interval translations and some other piecewise continuous maps
Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28,
Old Peterhof,
198504 St. Petersburg, Russia.
* Corresponding author: kryzhevicz@gmail.com
Received:
16
December
2018
Accepted:
14
September
2019
We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided all discontinuity points are wandering.
Mathematics Subject Classification: 37A05 / 37E05
Key words: Krylov-Bogolybov theorem / invariant measures / periodic points / piecewise continuous maps / piecewise isometries / interval translation maps
© EDP Sciences, 2020
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