PhD Oral Exam - A H M Mahbubur Rahman, Mathematics

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

This thesis focuses on - (1) the study of the existence and exactness of Absolutely Continuous Invariant Measures (acim) for piecewise convex maps with a countable number of branches, (2) developing the Ulam's method for approximation of density function and (3) the study of the existence of Absolutely Continuous Invariant Measures (acim) for piecewise concave maps using conjugation.

For the first topic, we investigate the existence and uniqueness of acim for two classes T_pc^∞ (I),T_pc^(∞,0) (I) of piecewise convex maps τ:I=[0,1]→[0,1] with countable number of branches. We provide sufficient conditions under which these maps have a unique acim, and we also give examples where multiple acims exist. Our results are based on the study of the Frobenius-Perron operator associated with the map, and we use analytical techniques to understand the properties of this operator.

The main result of this Ph.D. thesis is the generalization of the existence of absolutely continuous invariant measure for piecewise convex maps of an interval with countable number of branches. We consider two classes T_pc^∞ (I),T_pc^(∞,0) (I) of piecewise convex maps τ:I=[0,1]→[0,1] with countable number of branches. For the first class T_pc^∞ (I), we consider piecewise convex maps τ:I=[0,1]→[0,1] with countable number of branches and arbitrary countable number of limit points of partition points separated from 0. For the second class T_pc^(∞,0) (I), we consider piecewise convex maps τ:I=[0,1]→[0,1] with countable number of branches with the partition points converging to 0. In the thesis, we study absolutely continuous invariant measures (acim) of τ∈T_pc^∞ (I) and τ∈T_pc^(∞,0) (I) respectively. We also consider non-autonomous dynamical systems of maps in T_pc^∞ (I) or T_pc^(∞,0) (I) and study the existence of acims for the limit map. Moreover, we study exactness of τ∈T_pc^∞ (I) and τ∈T_pc^(∞,0) (I) respectively. We give a similar result for piecewise concave maps as well.

We also discuss the approximation of acims for piecewise convex maps with countable number of branches using Ulam's method. We also investigate the existence and uniqueness of ACIMs for two classes, T_pc^∞ (I) and T_pc^(∞,1) (I) of piecewise concave mapping σ. We use conjugation of piecewise convex map τ with countable number of branches that is defined in chapter 3 which implies that σ preserves a normalized absolutely continuous invariant measure whose density is an increasing function.