Hausdorff dimension of the level sets of self-affine functions

Abstract

Given a partition of [0,1] by closed intervals I1∪⋯∪Ik=[0,1] with k≥2. Given 0<α<1 and closed intervals Ji (i=1,⋯,k) with |Ij|=|Jj|α. Given τ1,⋯,τk∈{0,1}. Let Ω⊂[0,1]×[0,1] be the compact set satisfying thatΩ=⋃i=1k(φIi,1×φJi,τi)(Ω), where for an interval I=[a,b]⊂[0,1] and τ∈{−1,1}, φI,τ:[0,1]→I is the linear map such that φI,1(0)=a, φI,1(1)=b and φI,−1(0)=b, φI,−1(1)=a. Such Ω is a graph of a Borel function fΩ almost surely and is called a self-affine set of α-function type. We obtain the Hausdorff dimension of the level set Ωy={x; (x,y)∈Ω} in the case that λ∘fΩ−1 has a bounded density with respect to λ, where λ is the Lebesgue measure on [0,1].