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
Cet article a éte moissonné depuis la source EDP Sciences

Voir la notice de l'article

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. 
@article{10_1051_mmnp_2019041,
     author = {Sergey Kryzhevich},
     title = {Invariant measures for interval translations and some other piecewise continuous maps},
     journal = {Mathematical modelling of natural phenomena},
     eid = {15},
     year = {2020},
     volume = {15},
     doi = {10.1051/mmnp/2019041},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2019041/}
}
TY  - JOUR
AU  - Sergey Kryzhevich
TI  - Invariant measures for interval translations and some other piecewise continuous maps
JO  - Mathematical modelling of natural phenomena
PY  - 2020
VL  - 15
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2019041/
DO  - 10.1051/mmnp/2019041
LA  - en
ID  - 10_1051_mmnp_2019041
ER  - 
%0 Journal Article
%A Sergey Kryzhevich
%T Invariant measures for interval translations and some other piecewise continuous maps
%J Mathematical modelling of natural phenomena
%] 15
%D 2020
%V 15
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2019041/
%R 10.1051/mmnp/2019041
%G en
%F 10_1051_mmnp_2019041
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

[1] M. Boshernitzan, I. Kornfeld Interval translation mappings Ergodic Theory Dyn. Syst 1995 821 832

[2] H. Bruin Renormalization in a class of interval translation maps of d branches Dyn. Syst 2007 11 24

[3] H. Bruin, G. Clack Inducing and unique ergodicity of double rotations Discrete Contin. Dyn. Syst 2012 4133 4147

[4] H. Bruin, S. Troubetzkoy The Gauss map on a class of interval translation mappings Israel J. Math 2003 125 148

[5] J Buzzi Piecewise isometries have zero topological entropy Ergodic Theory Dyn. Syst 2001 1371 1377

[6] J. Buzzi, P Hubert Piecewise monotone maps without periodic points: Rigidity, measures and complexity Ergodic Theory Dyn. Syst 2004 383 405

[7] X.-C. Fu, J. Duan Global attractors and invariant measures for non-invertible planar piecewise isometric maps Phys. Lett. A 2007 285 290

[8] A Goetz Sofic subshifts and piecewise isometric systems Ergodic Theory Dyn. Syst 1999 1485 1501

[9] A Goetz Dynamics of piecewise isometries Illinois J. Math 2000 465 478

[10] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press (1997).

[11] B. Pires, Invariant measures for piecewise continuous maps. Preprint arXiv:1603.02542 (2016).

[12] J. Schmeling and S. Troubetzkoy, Interval Translation Mappings. In Dynamical systems (Luminy-Marseille) (1998) 291–302.

[13] H. Suzuki, S. Ito, K. Aihara Double rotations Discrete Contin. Dyn. Syst 2005 515 532

[14] S. Truong, Dynamics of piecewise translation maps. Preprint arXiv:1610.04700 (2017).

[15] M. Viana Ergodic theory of interval exchange maps Rev. Matem. Complut 2006 7 100

[16] D. Volk Almost every interval translation map of three intervals is finite type Discrete Continu. Dyn. Syst. A 2014 2307 2314

[17] D. Volk, Attractors of Piecewise Translation Maps. Preprint arXiv:1708.03780 (2017).

[18] J.-C. Yoccoz, Continued Fraction Algorithms for Interval Exchange Maps: an Introduction. Available on: https://www.college-de-france.fr/media/jean-christophe-yoccoz/UPL13956˙yoccoz.pdf (2020).

[19] C. Zhan-He, Y. Rong-Zhong, F. Xin-Chu Invariant measures for planar piecewise isometries J. Shanghai Univ. (Engl. Ed.) 2010 174 176

Cité par Sources :