TWO PROBLEMS IN DYNAMICS ON TORI

Author:

Liu, Qiao

Graduate Program:

Mathematics

Degree:

Doctor of Philosophy

Document Type:

Dissertation

Date of Defense:

February 09, 2021

Committee Members:

Federico Rodriguez Hertz, Major Field Member
Zhiren Wang, Chair & Dissertation Advisor
Martin Bojowald, Outside Unit & Field Member
Boris Kalinin, Major Field Member
Carina Curto, Program Head/Chair

Keywords:

Group action
Hausdorff dimension
Local rigidity
Dynamical system

Abstract:

In this dissertation, two problems about dynamics by toral automorphisms are investigated. In chapter 1 and 2, local rigidity property of affine $\mathbb Z \ltimes_\lambda \mathbb R$-action on tori generated by an irreducible toral automorphism and a linear flow along an eigenspace is studied. Such an action exhibits a weak version of local rigidity, i.e., any smooth perturbation close enough to an affine action is smoothly conjugate to the affine action up to a constant time change. In chapter 3 and 4, exceptional points whose orbits under an ergodic toral automorphism $A$ are sequentially $\epsilon$-concentrating on atoms are defined and studied, More precisely, we prove the Hausdorff dimension of the set \[ Z_{\epsilon, A}=\left \{x\in X \biggm | \begin{array}{l} \exists \{n_k\}_k\subset \mathbb N \text{ such that} \, \mu_x=\lim_{k\rightarrow \infty} \delta_x^{n_k} \, \text{exists} \\ \text{and} \, \mu_x \, \text{has at least} \, \epsilon \text{-portion supported on atoms.} \end{array} \right\} \] is at least $\kappa_{A}\epsilon$ where $\kappa_{A}$ only depends on $A$.