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

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

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

Jon Chaika ^1, and Howard Masur ^2,

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

Abstract
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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.

Keywords: generic points, Interval exchanges, Rauzy induction..

Mathematics Subject Classification: Primary: 37E0.

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:

[1]	A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5.
[2]	V. Cyr and B. Kra, Counting generic measures for a subshift of linear growth, arXiv:1505.02748.
[3]	A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778.
[4]	M. Keane, Interval exchange trasformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.
[5]	M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.
[6]	S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X.
[7]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.
[8]	E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 860-878.
[9]	M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621.
[10]	W. Veech, A Kronecker-Weyl theorem modulo 2, Proc. Nat. Acad. Sci. U.S.A., 60 (1968), 1163-1164. doi: 10.1073/pnas.60.4.1163.
[11]	W. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272. doi: 10.1007/BF02790174.
[12]	W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.
[13]	J.-C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1-69.

show all references

References:

[1]	A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211. doi: 10.1007/s10240-006-0001-5.
[2]	V. Cyr and B. Kra, Counting generic measures for a subshift of linear growth, arXiv:1505.02748.
[3]	A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778.
[4]	M. Keane, Interval exchange trasformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.
[5]	M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.
[6]	S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X.
[7]	H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.
[8]	E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 860-878.
[9]	M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621.
[10]	W. Veech, A Kronecker-Weyl theorem modulo 2, Proc. Nat. Acad. Sci. U.S.A., 60 (1968), 1163-1164. doi: 10.1073/pnas.60.4.1163.
[11]	W. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272. doi: 10.1007/BF02790174.
[12]	W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.
[13]	J.-C. Yoccoz, Interval exchange maps and translation surfaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1-69.

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