Journal of Modern Dynamics (JMD)

Growth and mixing

Pages: 315 - 338, Issue 2, April 2008      doi:10.3934/jmd.2008.2.315

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Krzysztof Frączek - Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland (email)
Leonid Polterovich - School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email)

Abstract: Given a bi-Lipschitz measure-preserving homeomorphism of a finite dimensional compact metric measure space, consider the sequence of the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin–Shapiro sequence

Keywords:  Growth rate of homeomorphism, the rate of mixing.
Mathematics Subject Classification:  37A05, 37A25, 37C05.

Received: August 2007;      Revised: November 2007;      Available Online: January 2008.