Dimensions of measures on carpets

Abstract

We aim to establish the Hausdorff dimensions for ergodic measures projected onto self-similar Sierpiński carpets, which are two-dimensional fractals given as invariant sets of iterated function systems. Our study considers a general case of overlapping on Sierpiński carpets by increasing a homogeneous contraction ratio, thereby failing the open set condition. Previous research
established the Hausdorff dimensions for self-similar and self-affine measures
defined on fat Sierpiński carpets for parameters in some range, which are discussed in the document. Our approach focuses on the disintegration of measures in R² into measures defined in R, the approximation of measures one-dimensional defined on sets akin to one-dimensional random fractals, and the exploitation of the exponential separation condition. Within the largest range of contraction ratios where the Hausdorff dimension of the disintegration of
a projected ergodic measure remains below or equal to 1, we determine the
Hausdorff dimension of the projected ergodic measure given as the entropy of
the measure over the Lyapunov exponent.
Furthermore, we focus on a specific class of Sierpiński carpets which are
given by IFS with a fixed contraction ratio. These carpets are parameterized by
a translation vector which permits horizontal overlapping based on a model by
Fraser and Shmerkin. However, the projection of carpets onto y-axis satisfies
the open set condition. We establish the Lq dimensions, q > 1, for self-similar
measures defined on this class of Sierpiński carpets.

Date of Award	29 Oct 2024
Original language	English
Awarding Institution	University of Bristol
Supervisor	Francesco Mezzadri (Supervisor) & Thomas M Jordan (Supervisor)