Dynamics of chessboard-style flow and piecewise linear circle homeomorphisms

Abstract

In this thesis, we provide an overview of the various dynamical properties of piecewise linear and piecewise smooth circle maps. We establish the existence of conjugation to rigid rational rotations and self-similarity structure in a family of piece-wise linear circle homeomorphisms induced by a certain type of piece-wise linear flow on R2, which we call the chessboard style flow. When such conjugation happens at a parameter, we call these special parameters rigid rational rotation points. We also proved various properties of the set of rigid rational rotation points, including self-similar and Cantor set-like structure. These results are used later in the thesis to prove a local linear scaling result of rotation numbers at some rigid rational rotation points when considered as a function of the parameters. The key method used to prove the main results in this thesis is the renormalisation algorithm. In contrast to the conventional renormalisation algorithm, which is based on continued fractions and commuting pairs, we developed a modified version which helps to reveal symmetries hidden within the family of circle homeomorphisms considered. We also provide explicit computations of the renormalisation function in a suitable coordinate system, which allows us to prove the main results later.

Date of Award	7 Jul 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Paul Glendinning (Co Supervisor) & James Montaldi (Main Supervisor)