Hausdorff dimension of the level sets of self-affine functions

Given a partition of [0, 1] by closed intervals I-1 boolean OR ... boolean OR I-k = [0,1] with k >= 2. Given 0 < alpha < 1 and closed intervals J(i) (i = 1, ... ,k) with vertical bar I-j vertical bar = vertical bar J(j)vertical bar(alpha). Given tau(1), ... ,tau(k) is an element of {0,1}. Let Omega subset of [0,1] x [0,1] be the compact set satisfying that Omega = U-i=1(k)(phi I-i,1 x phi J(i),tau(i))(Omega) where for an interval I = [a, b] subset of [0, 1] and tau is an element of {-1, 1}, phi(I,tau) : [0,1] -> I is the linear map such that phi(I,1)(0)= a, phi(I,,1)(1) = b and phi(I,,-1) (0)= b, phi(I,,-1) (1) = a. Such Omega is a graph of a Sorel function f(Omega) almost surely and is called a self-affine set of alpha-function type. We obtain the Hausdorff dimension of the level set Omega(y) = {x; (x, y) is an element of Omega} in the case that lambda o f(Omega)(-1) has a bounded density with respect to lambda, where lambda is the Lebesgue measure on [0, 1]. (C) 2014 Elsevier Inc. All rights reserved.