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

We investigate Takagi-type functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H=frac{loggamma}{log eh}.$ 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 show that the SBR measure is absolutely continuous for large enough $gamma$. Dually, where duality is related to time reversal, we prove that for large enough $gamma$ a version of the Takagi-type curve centered around fibers of the associated stable manifold possesses a square integrable local time.