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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

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.  Google Scholar

[2]

V. Cyr and B. Kra, Counting generic measures for a subshift of linear growth,, , ().   Google Scholar

[3]

A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778.  Google Scholar

[4]

M. Keane, Interval exchange trasformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.  Google Scholar

[5]

M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.  Google Scholar

[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.  Google Scholar

[7]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.  Google Scholar

[8]

E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 860-878.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272. doi: 10.1007/BF02790174.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. Cyr and B. Kra, Counting generic measures for a subshift of linear growth,, , ().   Google Scholar

[3]

A. B. Katok, Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, 211 (1973), 775-778.  Google Scholar

[4]

M. Keane, Interval exchange trasformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.  Google Scholar

[5]

M. Keane, Non-ergodic interval exchange transformations, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.  Google Scholar

[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.  Google Scholar

[7]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200. doi: 10.2307/1971341.  Google Scholar

[8]

E. A. Sataev, The number of invariant measures for flows on orientable surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 860-878.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272. doi: 10.1007/BF02790174.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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