Dynamics of Non-Classical Interval Exchanges

Citation

Gadre, Vaibhav S. (2010) Dynamics of Non-Classical Interval Exchanges. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/082X-F229. https://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.

Thesis Files