Dimension and measure of baker-like skew-products of $β$-transformations

Abstract

We consider a generalisation of the baker's transformation, consisting of a skew-product of contractions and a $β$-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.

14 pages, 3 figures. Some typos have been corrected, and some arguments have been written out in greater details