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 $γ$ that are Hölder continuous with coefficient $H=\frac{\logγ}{\log \frac{1}{2}}.$ 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 $γ= 2^{-\frac{1}{m}}, m\in\mathbb{N},$ that the Takagi-type curves possess square integrable local times with $m-2$ smooth derivatives.

Figures (1)

• Figure 1: Takagi curve for $\gamma=2^{-\frac{1}{2}}$, $\gamma=2^{-\frac{1}{3}}$, $\gamma=2^{-\frac{1}{6}}$, $\gamma=2^{-\frac{1}{10}}$

Theorems & Definitions (32)

• Proposition 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Proposition 3.4
• Lemma 3.5
• Corollary 3.6