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Mild mixing of certain interval-exchange transformations

Published online by Cambridge University Press:  18 May 2017

DONALD ROBERTSON
DONALD ROBERTSON*
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA email [email protected]
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Abstract

We prove that irreducible, linearly recurrent, type W interval-exchange transformations are always mild mixing. For every irreducible permutation, the set of linearly recurrent interval-exchange transformations has full Hausdorff dimension.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 1 , January 2019 , pp. 248 - 256
Copyright
© Cambridge University Press, 2017 

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References

Avila, A. and Forni, G.. Weak mixing for interval exchange transformations and translation flows. Ann. of Math. (2) 165(2) (2007), 637–664.Google Scholar
Bergelson, V., del Junco, A., Lemańczyk, M. and Rosenblatt, J.. Rigidity and non-recurrence along sequences. Ergod. Th. & Dynam. Sys. 34(5) (2014), 1464–1502.Google Scholar
Boshernitzan, M. and Nogueira, A.. Generalized eigenfunctions of interval exchange maps. Ergod. Th. & Dynam. Sys. 24(3) (2004), 697–705.Google Scholar
Boshernitzan, M. D.. Rank two interval exchange transformations. Ergod. Th. & Dynam. Sys. 8(3) (1988), 379–394.Google Scholar
Chaika, J., Cheung, Y. and Masur, H.. Winning games for bounded geodesics in moduli spaces of quadratic differentials. J. Mod. Dyn. 7(3) (2013), 395–427.Google Scholar
Chaika, J., Damanik, D. and Krüger, H.. Schrödinger operators defined by interval-exchange transformations. J. Mod. Dyn. 3(2) (2009), 253–270.Google Scholar
Chaves, J. and Nogueira, A.. Spectral properties of interval exchange maps. Monatsh. Math. 134(2) (2001), 89–102.Google Scholar
Ferenczi, S., Holton, C. and Zamboni, L. Q.. Joinings of three-interval exchange transformations. Ergod. Th. & Dynam. Sys. 25(2) (2005), 483–502.Google Scholar
Fraczek, K. and Lemańczyk, M.. On mild mixing of special flows over irrational rotations under piecewise smooth functions. Ergod. Th. & Dynam. Sys. 26(3) (2006), 719–738.Google Scholar
Fraczek, K., Lemańczyk, M. and Lesigne, E.. Mild mixing property for special flows under piecewise constant functions. Discrete Contin. Dyn. Syst. 19(4) (2007), 691–710.Google Scholar
Fraczek, K.. Density of mild mixing property for vertical flows of abelian differentials. Proc. Amer. Math. Soc. 137(12) (2009), 4129–4142.Google Scholar
Hmili, H.. Non topologically weakly mixing interval exchanges. Discrete Contin. Dyn. Syst. 27(3) (2010), 1079–1091.Google Scholar
Katok, A.. Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35(4) (1980), 301–310.Google Scholar
Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.Google Scholar
Kanigowski, A. and Kułaga-Przymus, J.. Ratner’s property and mild mixing for smooth flows on surfaces. Ergod. Th. & Dynam. Sys. 36 (2016), 2512–2537.Google Scholar
Kleinbock, D. and Weiss, B.. Bounded geodesics in moduli space. Int. Math. Res. Not. IMRN 30 (2004), 1551–1560.Google Scholar
Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115(1) (1982), 169–200.Google Scholar
Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115(1) (1982), 201–242.Google Scholar
Veech, W. A.. The metric theory of interval exchange transformations. I. Generic spectral properties. Amer. J. Math. 106(6) (1984), 1331–1359.Google Scholar
Walters, P.. Some invariant 𝜎-algebras for measure-preserving transformations. Trans. Amer. Math. Soc. 163 (1972), 357–368.Google Scholar