On the intersection of Cantor sets and products of random matrices

TL;DR

This paper develops new methods to compute the Hausdorff dimension of intersections of Cantor sets and their translates by analyzing the Lyapunov exponent of products of random matrices, especially in cases with continuous stationary measures.

Contribution

It introduces novel combinatorial and analytic tools to compute Lyapunov exponents for a broad class of examples with continuous stationary measures, extending previous explicit computations.

Findings

Computed Hausdorff dimension for intersections with forbidden digits

Determined dimension of middle-seventh Cantor set with random translates

Extended dimension calculations to cases with continuous stationary measures

Abstract

Kenyon and Peres (1991) showed that the Hausdorff dimension of intersections of randomly translated Cantor sets can be expressed in terms of the top Lyapunov exponent of a product of random matrices, and this exponent can be written as an integral with respect to stationary measures on the projective line. Although explicit computations are available when stationary measures are discrete, the continuous case has remained challenging. In this paper we introduce new combinatorial and analytic tools that allow us to compute the Lyapunov exponent, and hence the Hausdorff dimension, in a broad class of examples where stationary measures are continuous. As an application, we complete the dimension computation in the setting where a single digit is forbidden; for example, we determine the Hausdorff dimension of the intersection of the middle-seventh Cantor set with a random translate of itself.