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January  2009, 3(1): 35-49. doi: 10.3934/jmd.2009.3.35

Weak mixing for logarithmic flows over interval exchange transformations

Corinna Ulcigrai 1,

1. 

School of Mathematics, University of Bristol, Bristol, United Kingdom

Received  August 2008 Revised  November 2008 Published  February 2009

We consider a class of special flows over interval exchange transformations which includes roof functions with symmetric logarithmic singularities. We prove that such flows are typically weakly mixing. As a corollary, minimal flows given by multivalued Hamiltonians on higher-genus surfaces are typically weakly mixing.
Keywords: logarithmic singularities, area-preserving flows on surfaces., Weak mixing, interval exchange transformations.
Mathematics Subject Classification: Primary: 37E35, Secondary: 37A2.
Citation: Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
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Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

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