# American Institute of Mathematical Sciences

2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289

## There exists an interval exchange with a non-ergodic generic measure

 1 Department of Mathematics, University of Utah, 155 S. 1400 E., Room 233, Salt Lake City, UT 84112 2 Department of Mathematics, University of Chicago, 5734 S. University Avenue, Room 208C, Chicago, IL 60637, United States

Received  November 2014 Revised  July 2015 Published  October 2015

We prove that there exists an interval exchange transformation and a point so that the orbit of the point equidistributes according to a non-ergodic measure. That is, it is possible for a non-ergodic measure to arise from the Krylov-Bogolyubov construction of invariant measures for an interval exchange transformation.
Citation: Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
##### References:

show all references

##### References:
 [1] Daniel Bernazzani. Most interval exchanges have no roots. Journal of Modern Dynamics, 2017, 11: 249-262. doi: 10.3934/jmd.2017011 [2] Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079 [3] David Ralston, Serge Troubetzkoy. Ergodicity of certain cocycles over certain interval exchanges. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2523-2529. doi: 10.3934/dcds.2013.33.2523 [4] Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143-153. doi: 10.3934/jmd.2017007 [5] Tomas Persson. Typical points and families of expanding interval mappings. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4019-4034. doi: 10.3934/dcds.2017170 [6] Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103 [7] Thomas Dauer, Marlies Gerber. Generic absence of finite blocking for interior points of Birkhoff billiards. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4871-4893. doi: 10.3934/dcds.2016010 [8] Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877 [9] Charles E. M. Pearce, Krzysztof Szajowski, Mitsushi Tamaki. Duration problem with multiple exchanges. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 333-355. doi: 10.3934/naco.2012.2.333 [10] Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209 [11] Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3433-3457. doi: 10.3934/dcds.2012.32.3433 [12] Jon Fickenscher. A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1983-2025. doi: 10.3934/dcds.2016.36.1983 [13] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2729-2749. doi: 10.3934/dcdss.2020457 [14] Marta Marulli, Vuk Miliši$\grave{\rm{c}}$, Nicolas Vauchelet. Reduction of a model for sodium exchanges in kidney nephron. Networks & Heterogeneous Media, 2021, 16 (4) : 609-636. doi: 10.3934/nhm.2021020 [15] Qixiang Wen, Shenquan Liu, Bo Lu. Firing patterns and bifurcation analysis of neurons under electromagnetic induction. Electronic Research Archive, , () : -. doi: 10.3934/era.2021034 [16] Dong Sun, V. S. Manoranjan, Hong-Ming Yin. Numerical solutions for a coupled parabolic equations arising induction heating processes. Conference Publications, 2007, 2007 (Special) : 956-964. doi: 10.3934/proc.2007.2007.956 [17] Bogdan Kwiatkowski, Tadeusz Kwater, Anna Koziorowska. Influence of the distribution component of the magnetic induction vector on rupturing capacity of vacuum switches. Conference Publications, 2011, 2011 (Special) : 913-921. doi: 10.3934/proc.2011.2011.913 [18] Will Brian, Jonathan Meddaugh, Brian Raines. Shadowing is generic on dendrites. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2211-2220. doi: 10.3934/dcdss.2019142 [19] Serge Troubetzkoy. Recurrence in generic staircases. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047 [20] Neal Koblitz, Alfred Menezes. Another look at generic groups. Advances in Mathematics of Communications, 2007, 1 (1) : 13-28. doi: 10.3934/amc.2007.1.13

2020 Impact Factor: 0.848