Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence and smoothness of local time
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2021
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| author | Imkeller, P. Pamen, O. Menoukeu |
| author_facet | Imkeller, P. Pamen, O. Menoukeu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_07185 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence and smoothness of local time Imkeller, P. Pamen, O. Menoukeu Dynamical Systems Probability primary 26A16, 37D20, 28A80, 60J55, secondary 28D05, 37C70, 37D10, 37H15, 42A55 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. |
| title | Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence and smoothness of local time |
| topic | Dynamical Systems Probability primary 26A16, 37D20, 28A80, 60J55, secondary 28D05, 37C70, 37D10, 37H15, 42A55 |
| url | https://arxiv.org/abs/2107.07185 |