Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence and smoothness of local time
We investigate the occupation measures and local times of Takagi-type functions with roughness parameter $\gamma$, which are H\"older continuous with exponent $H=\frac{\log\gamma}{\log(1/2)}.$ Analytical insight is obtained by embedding these functions into a dynamical system related to the baker transform, whose global attractor is the graph of the Takagi function. The associated stable manifolds support Sinai-Bowen-Ruelle (SBR) measures, which we identify with the laws of certain symmetric Bernoulli convolutions. Dually, where duality is induced by time reversal, we derive a representation of the Takagi-type curves centred around the stable fibres in terms of Bernoulli convolutions, thereby relating SBR measures to occupation measures. While Bernoulli convolutions belong to the first Rademacher chaos, we show that the occupation measure is naturally represented as a functional in the second Rademacher chaos within the framework of non-Gaussian Malliavin calculus. Using a Fourier-analytic criterion together with variants of Weyl's equidistribution theorem, we prove that Takagi-type curves admit square-integrable local times for $\gamma=2^{-1/m}, \, m\geq 7,$ and that the same conclusion holds for drifted Takagi curves for almost every $\gamma\in(1/2,1)$.