An entropy formula for a non-self-affine measure with application to Weierstrass-type functions

Abstract

Let $ \tau : [0,1] \rightarrow [0,1] $ be a piecewise expanding map with full branches. Given $ \lambda : [0,1] \rightarrow (0,1) $ and $ g : [0,1] \rightarrow \mathbb{R} $ satisfying $ \tau ' \lambda > 1 $, we study the Weierstrass-type function \[ \sum _{n=0} ^\infty \lambda ^n (x) \, g (\tau ^n (x)), \] where $ \lambda ^n (x) := \lambda(x) \lambda (\tau (x)) \cdots \lambda (\tau ^{n-1} (x)) $. Under certain conditions, Bedford proved that the box counting dimension of its graph is given as the unique zero of the topological pressure function \[ s \mapsto P ((1-s) \log \tau ' + \log \lambda) . \] We give a sufficient condition under which the Hausdorff dimension also coincides with this value. We adopt a dynamical system theoretic approach which was originally used to investigate special cases including the classical Weierstrass functions. For this purpose we prove a new Ledrappier-Young entropy formula, which is a conditional version of Pesin's formula, for non-invertible dynamical systems. Our formula holds for all lifted Gibbs measures on the graph of the above function, which are generally not self-affine.