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arXiv · 0906.2563

Dynamics of Non-Classical Interval Exchanges

Vaibhav S Gadre
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Abstract

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orientable flat surfaces. Non-classical interval exchanges can be studied as a dynamical system by considering Rauzy induction in this context. This gives a refinement process on the parameter space similar to Kerckhoff's simplicial systems. We show that the refinement process gives an expansion that has a key dynamical property called {\it uniform distortion}. We use uniform distortion to prove normality of the expansion. Consequently, we prove an analog of Keane's conjecture: almost every non-classical interval exchange is uniquely ergodic. Uniform distortion has been independently shown by Avila-Resende.

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Vaibhav S Gadre. 2009-06-14. Dynamics of Non-Classical Interval Exchanges. https://arxiv.org/abs/0906.2563

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