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Dynamics of non-classical interval exchanges
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Gadre, Vaibhav (2012) Dynamics of non-classical interval exchanges. Ergodic Theory and Dynamical Systems, Vol.32 (No.6). pp. 1930-1971. doi:10.1017/S0143385711000691 ISSN 0143-3857.
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Official URL: http://dx.doi.org/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: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Ergodic Theory and Dynamical Systems | ||||
Publisher: | Cambridge University Press | ||||
ISSN: | 0143-3857 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Vol.32 | ||||
Number: | No.6 | ||||
Page Range: | pp. 1930-1971 | ||||
DOI: | 10.1017/S0143385711000691 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
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