Dimension and measure of baker-like skew-products of $\beta$-transformations

TL;DR

This paper studies a generalized baker-like skew-product involving contractions and a beta-transformation, calculating the Hausdorff dimension and Lebesgue measure of its attractor for a positive measure set of parameters using a novel transversality lemma.

Contribution

It introduces a new transversality lemma applicable to a broader parameter set, enabling the calculation of dimension and measure for this class of skew-products.

Findings

Hausdorff dimension of attractor computed for positive measure parameter set

Lebesgue measure of attractor determined for the same parameter set

New transversality lemma extends Solomyak's approach to a larger class of maps

Abstract

We consider a generalisation of the baker's transformation, consisting of a skew-product of contractions and a $\beta$-transformation. The Hausdorff dimension and Lebesgue measure of the attractor is calculated for a set of parameters with positive measure. The proofs use a new transverality lemma similar to Solomyak's [Solomyak, 1995]. This transversality, which is applicable to the considered class of maps holds for a larger set of parameters than Solomyak's transversality.