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Dynamics of non-classical interval exchanges

Citation

Gadre, Vaibhav S. (2010) Dynamics of non-classical interval exchanges. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:04212010-181932550

Abstract

Train tracks with a single vertex are a generalization of interval exchange maps. Here, we consider non-classical interval exchanges: complete train tracks with a single vertex. These 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. In the concluding chapter, we state an application of the main results of the thesis to a question about harmonic measures on the Thurston boundary of Teichmuller space coming from finitely supported random walks on the mapping class group.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Interval exchange, Teichmuller space, Mapping class group.
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Dunfield, Nathan M. (co-advisor)
  • Calegari, Danny C. (co-advisor)
Thesis Committee:
  • Dunfield, Nathan M.
  • Calegari, Danny C. (chair)
  • Ramakrishnan, Dinakar
  • Ni, Yi
Defense Date:2 April 2010
Author Email:vaibhav (AT) caltech.edu
Record Number:CaltechTHESIS:04212010-181932550
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:04212010-181932550
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5731
Collection:CaltechTHESIS
Deposited By: Vaibhav Gadre
Deposited On:21 May 2010 15:48
Last Modified:23 Sep 2014 17:17

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