The Hausdorff dimension of some random invariant graphs

Weierstrass' example of an everywhere continuous but nowhere differentiable function is given by w(x) = σ n=0 ∞ λ n cos 2πb nx where λ ∈ (0, 1), b ≥ 2, λb > 1. There is a well-known and widely accepted, but as yet unproven, formula for the Hausdorff dimension of the graph of w. Hunt [H] proved that this formula holds almost surely on the addition of a random phase shift. The graphs of Weierstrass-type functions appear as repellers for a certain class of dynamical system; in this paper we prove formulae analogous to those for random phase shifts of w(x) but in a dynamic context. Let T : S 1 → S 1 be a uniformly expanding map of the circle. Let λ : S 1 → (0, 1), p : S 1 → ℝ and define the function w(x) = σ n=0 ∞ λ(x)λ(T (x)) · · · λ(T n-1(x))p(T n(x)). The graph of w is a repelling invariant set for the skew-product transformation T (x, y) = (T (x), λ(x) -1(y-p(x))) on S 1 × ℝ and is continuous but typically nowhere differentiable. With the addition of a random phase shift in p, and under suitable hypotheses including a partial hyperbolicity assumption on the skew-product, we prove an almost sure formula for the Hausdorff dimension of the graph of w using a generalization of techniques from [H] coupled with thermodynamic formalism. © 2012 IOP Publishing Ltd & London Mathematical Society.