Dimension theory on projections of self-conformal and self-affine sets

Abstract

In this thesis we study strong projection theorems. That is, results which show that the orthogonal projection of a certain set or measure is the same and has the maximum possible value in every direction. Such a theorem is a strengthening of a Marstrand-type projection theorem, true for almost all projections. We examine this in the context of Borel probability measures on dynamically-defined fractal sets. We also obtain projection results on the sets themselves as corollaries of the theorems for measures. In order to satisfy a strong projection theorem, these sets are subject to certain minimality assumptions relating to the underlying iterated function system. Our first aim is to generalise a strong projection theorem of Hochman and Shmerkin on self-similar sets and measures with dense rotations. Our result holds for Gibbs measures on self-conformal sets in d-dimensional Euclidean space satisfying certain minimal assumptions, without requiring any separation condition. Via a strong variational principle we obtain a strong projection theorem for these self-conformal sets. As a corollary we show that Falconer’s distance set conjecture holds for this class of self-conformal sets satisfying the open set condition. Following this we extend a projection theorem of Hochman and Shmerkin on products of measures invariant under multiplicatively independent dynamics. We introduce randomness into the setting in the form of random cascade measures on T_a, T_b-invariant sets for a, b multiplicatively independent natural numbers. We extend the strong projection theorem in the case when the iterated function systems underlying the canonical mappings satisfy the open set condition. This yields a stochastic version of Furstenberg's sumset conjecture. We also achieve a lower bound for the dimension of projections of products of more general self-similar sets. In this case a strong projection theorem holds when the sets have equal similarity and Hausdorff dimensions.

Date of Award	31 Dec 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Charles Walkden (Supervisor) & Xiong Jin (Supervisor)