Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence and smoothness of local time

Abstract

We investigate Takagi-type functions with roughness parameter $\gamma$ that are H\"older continuous with coefficient $H=\frac{\log\gamma}{\log \frac{1}{2}}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We identify these measures with the laws of certain symmetric Bernoulli convolutions. Dually, where duality is related to ''time'' reversal, we give a representation of the Takagi-type curves centered around fibers of the associated stable manifold in terms of Bernoulli convolutions. Duality also relates SBR to occupation measure. As opposed to SBR measure - Bernoulli convolutions belong to the first chaos - occupation measure turns out to be a functional in the second Rademacher chaos, in terms of this non-Gaussian Malliavin calculus. Using a Fourier analytic criterion and variants of Weyl's equidistribution theorem, we prove for smoothness parameters $\gamma = 2^{-\frac{1}{m}}, m\in\mathbb{N},$ that the Takagi-type curves possess square integrable local times with $m-2$ smooth derivatives.