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

TL;DR

This paper explores Takagi-type functions through dynamical systems, revealing their measure-theoretic properties, smoothness, and local times, with connections to Bernoulli convolutions and non-Gaussian Malliavin calculus.

Contribution

It establishes a novel dynamical systems framework for analyzing Takagi-type functions, linking their measures to Bernoulli convolutions and proving smoothness of their local times for specific parameters.

Findings

Takagi-type functions are embedded into dynamical systems related to the baker transform.

SBR measures are identified with laws of symmetric Bernoulli convolutions.

For certain smoothness parameters, the functions have square integrable local times with smooth derivatives.

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.