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

  • Catherine Bruce

Student thesis: Phd


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 Award31 Dec 2023
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorCharles Walkden (Supervisor) & Xiong Jin (Supervisor)


  • percolations
  • self-affine sets
  • CP-chains
  • self-conformal sets
  • Hausdorff dimension
  • fractals
  • orthogonal projections

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