Dynamics of non-classical interval exchanges

Gadre, V. H. (2012) Dynamics of non-classical interval exchanges. Ergodic Theory and Dynamical Systems, 32(6), pp. 1930-1971. (doi: 10.1017/S0143385711000691)

Abstract

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira [Measured foliations on non-orientable surfaces. Ann. Sci. Éc. Norm. Supér. (4) 26(6) (1993), 645–664]. 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 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 in [A. Avila and M. Resende. Exponential mixing for the Teichmüller flow in the space of quadratic differentials, http://arxiv.org/abs/0908.1102].

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Gadre, Dr Vaibhav
• Authors: Gadre, V. H.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Ergodic Theory and Dynamical Systems
• Publisher: Cambridge University Press
• ISSN: 0143-3857
• ISSN (Online): 1469-4417
• Copyright Holders: Copyright © 2012 Cambridge University Press
• First Published: First published in Ergodic Theory and Dynamical Systems 32(6):1930-1971
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.