2018/12/11 by Sergey Kryzhevich, Kryzhevich, Sergey
Mathematics · #37A05 #37E05 #37E10 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1812.04534
openalex publication_date 2018/12/11 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
We study some special classes of piecewise continuous maps on a finite smooth\npartition of a compact manifold and look for invariant measures for such maps.\nWe show that in the simplest one-dimensional case (so-called interval\ntranslation maps) a Borel probability non-atomic invariant measure exists for\nany map. We use this result to demonstrate that any interval translation map\nendowed with such a measure is metrically equivalent to an interval exchange\nmap. Finally, we study the general case of piecewise continuous maps and prove\na simple result on existence of an invariant measure provided all discontinuity\npoints are wandering.\n