Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T17:42:59.976Z Has data issue: false hasContentIssue false
  • English
  • Français

There exists a topologically mixing interval exchange transformation

Published online by Cambridge University Press:  24 May 2011

JON CHAIKA
JON CHAIKA*
Affiliation:
Department of Mathematics, Rice University, Houston, TX 77005, USA (email: jonchaika@math.uchicago.edu, Jonathan.M.Chaika@rice.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the existence of a topologically mixing interval exchange transformation and prove that no interval exchange is topologically mixing of all orders.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 869 - 875
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Apostol, T.. Introduction to Analytic Number Theory. Springer, Berlin, 1976.Google Scholar
[2]Avila, A. and Forni, G.. Weak mixing for interval exchange transformations and translation flows. Ann. of Math. (2) 165(2) (2007), 637–664.CrossRefGoogle Scholar
[3]Boshernitzan, M. and Chaika, J.. Diophantine properties of IETs and general systems: quantitative proximality and connectivity. Preprint, arXiv: 0910.5422.Google Scholar
[4]Boshernitzan, M. and Nogueira, A.. Generalized eigenfunctions of interval exchange maps. Ergod. Th. & Dynam. Sys. 24(3) (2004), 697–705.CrossRefGoogle Scholar
[5]Chaika, J., Damanik, D. and Krueger, H.. Schrodinger operators defined by interval exchange transformations. J. Mod. Dyn. 3(2) (2009), 253–270.Google Scholar
[6]Chaves, J. and Nogueira, A.. Spectral properties of interval exchange maps. Monatsh. Math. 134(2) (2001), 89–102.CrossRefGoogle Scholar
[7]Dekking, F. M. and Keane, M.. Mixing properties of substitutions. Z. Wahrschein. Verw. Gebiete 42(1) (1978), 23–33.CrossRefGoogle Scholar
[8]Ferenczi, S.. Rank and symbolic complexity. Ergod. Th. & Dynam. Sys. 16(4) (1996), 663–682.CrossRefGoogle Scholar
[9]Katok, A.. Interval exchange transformations and some special flows are not mixing. Israel J. Math 35(4) (1980), 301–310.CrossRefGoogle Scholar
[10]Katok, A. B. and Stepin, A. M.. Approximations in ergodic theory. Uspekhi Mat. Nauk 22(5(137)) (1967), 81–106 (in Russian).Google Scholar
[11]Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26(2) (1977), 188–196.CrossRefGoogle Scholar
[12]Nogueira, A. and Rudolph, D.. Topological weak-mixing of interval exchange maps. Ergod. Th. & Dynam. Sys. 17 (1997), 1183–1209.CrossRefGoogle Scholar
[13]Veech, W.. The Metric Theory of interval exchange transformations I. Amer. J. Math. 106(6) (1984), 1331–1359.CrossRefGoogle Scholar