Geodesic
Invariant measures for interval translations and some other piecewise continuous maps
Sergey Kryzhevich1
1Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28, Old Peterhof, 198504 St. Petersburg, Russia.
Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 15

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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.
DOI : 10.1051/mmnp/2019041

Sergey Kryzhevich  1

1 Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskij pr. 28, Old Peterhof, 198504 St. Petersburg, Russia. 
Sergey Kryzhevich. Invariant measures for interval translations and some other piecewise continuous maps. Mathematical modelling of natural phenomena, Tome 15 (2020), article  no. 15. doi: 10.1051/mmnp/2019041
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