Dissertations, Theses, and Capstone Projects

Date of Degree

9-2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy

Program

Mathematics

Advisor

Enrique Pujals

Committee Members

Michael Shub

Ara Basmajian

Subject Categories

Dynamical Systems | Geometry and Topology

Keywords

Topological entropy, homology, intersections

Abstract

A central line of research in dynamical systems concerns the identification of geometric mechanisms necessary and/or sufficient for the production of entropy. In this thesis we prove several technical results directed (in some cases rather obliquely) towards this general project, motivated in particular by problems arising in attempts to prove Shub's entropy conjecture in finite regularity. We first prove a generalization of Poincaré's formula in integral geometry to arbitrary closed manifolds, where the transformation group is replaced by a compact family of C^1 diffeomorphisms satisfying two (essentially optimal) transversality conditions. We give two dynamical applications of this result, first a stochastic Artin-Mazur bound for arbitrary C^1 diffeomorphisms in the presence of sufficiently rich random noise, and second to the construction of uniformly expanding measures on Diff(M). Moving to the C^2 setting, we investigate domination assumptions necessary to guarantee the flatness of Bowen disks on subexponentially shrinking scales. Finally, we answer a question of Correa \& Pujals by constructing a family of subshifts (and smooth realizations) with superpolynomial and subexponential generalized entropy. Returning to the entropy conjecture, we conclude by sketching some directions for future work.

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