PhD Oral Exam - Aparna Rajput, Mathematics and Statistics

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

The main objective of this thesis proposal is to prove the quasi-compactness of the Frobenius-Perron operator for two distinct types of interval maps. In the first part, we focus on piecewise convex maps with an infinite number of branches defined on the unit interval [0, 1]. We show that for sufficiently high iterates, these maps exhibit piecewise expanding behavior. We prove the Lasota-Yorke inequality for the pair (BV, L¹) and, using the Ionescu Tulcea-Marinescu theorem, we establish the existence of an absolutely continuous invariant measure (ACIM) and the quasi-compactness of the Frobenius-Perron operator associated with these maps, revealing a range of strong ergodic properties for the system. Additionally, we demonstrate the exactness of the dynamical system.

In the second part of this thesis, we investigate piecewise expanding C^(1+ε) interval maps. By proving a Lasota-Yorke inequality for the Frobenius-Perron operator for appropriate spaces and again using the Ionescu Tulcea-Marinescu theorem, we show the existence of an ACIM. Furthermore, we establish the quasi-compactness of the Frobenius-Perron operator for these maps and explore key dynamical properties of the system, such as weak mixing and exponential decay of correlations.