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arXiv · 1503.06451

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

Atsuya Otani
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Abstract

Let $ τ: [0,1] \rightarrow [0,1] $ be a piecewise expanding map with full branches. Given $ λ: [0,1] \rightarrow (0,1) $ and $ g : [0,1] \rightarrow \mathbb{R} $ satisfying $ τ' λ> 1 $, we study the Weierstrass-type function \[ \sum _{n=0} ^\infty λ^n (x) \, g (τ^n (x)), \] where $ λ^n (x) := λ(x) λ(τ(x)) \cdots λ(τ^{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 \[ s \mapsto P ((1-s) \log τ' + \log λ) . \] 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.

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Atsuya Otani. 2015-07-14. An entropy formula for a non-self-affine measure with application to Weierstrass-type functions. https://arxiv.org/abs/1503.06451

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