Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of R^k
Open Access
Author:
Uzman, Ali Alp
Graduate Program:
Mathematics (PHD)
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
February 24, 2023
Committee Members:
Federico Rodriguez Hertz, Chair & Dissertation Advisor Zhiren Wang, Major Field Member Alexei Novikov, Professor in Charge/Director of Graduate Studies Martin Bojowald, Outside Unit & Field Member Boris Kalinin, Major Field Member
Keywords:
arithmeticity measure rigidity smooth ergodic theory dynamical systems
Abstract:
We prove an arithmeticity theorem in the context of nonuniform measure rigidity. Adapting machinery developed by A. Katok and F. Rodriguez Hertz [J. Mod. Dyn. 10 (2016), 135–172; MR3503686] for Z^k systems to R^k systems, we show that any maximal rank positive entropy system on a manifold generated by k>=2 commuting vector fields of regularity C^r for r>1 is measure theoretically isomorphic to a constant time change of the suspension of some action of Z^k on the (k+1)-torus or the (k+1)-torus modulo {id,-id} by affine automorphisms with linear parts hyperbolic. Further, the constructed conjugacy has certain smoothness properties. This in particular answers a problem and a conjecture from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin [Ann. of Math. (2) 174 (2011), no. 1, 361–400; MR2811602].
