Two-dimensional random substitutions and their associated dynamical systems

PhD Dissertation Defense
Two-dimensional random substitutions and their associated dynamical systems

by Bryan Ceasar L. Felipe
PhD Mathematics Candidate

Date: Saturday, 13 August 2022
Time: 5 pm
Venue: Online

Advisers:
Dr. Eden Delight P. Miro
Ateneo de Manila University

Panelists:
Dr. Alvaro Bustos-Gajardo
The Open University
Dr. Nathan V. Caalim
University of the Philippines Diliman
Dr. Job A. Nable
Ateneo de Manila University
Dr. Gwendolyn S. Tadeo
Saint Louis University
 

This study presents a generalization of random substitutions to two dimensions. We define the rectangular-preserving random substitutions and digit random substitutions which are generalizations of their deterministic counterparts. We then discuss the associated two-dimensional subshifts to these substitutions and present some dynamical properties of these systems by generalizing previous results on one-dimensional random substitutions. For rectangular-preserving random substitutions, we also investigate the existence of periodic points in their associated subshifts. We give a necessary criterion for primitive compatible rectangular-preserving random substitutions to admit periodic points in their subshifts. A stronger criterion is provided for the case of block random substitutions. Moreover, we present an algorithm that determines if a rectangular word is a valid periodic block for a periodic point in a rectangular-preserving random substitution subshift.

Key Words: random substitution, rectangular-preserving substitution, digit substitution, periodic points, symbolic dynamics