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

arXiv2107.07185MaRDI QIDQ6505149

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Abstract: We investigate Takagi-type functions with roughness parameter $gamma$ that are H"older continuous with coefficient $H=fracloggammalogeh.$ 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.

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