Scaling properties of the Thue–Morse measure

M. Baake; P. Gohlke; M. Kesseböhmer; T. Schindler

doi:10.3934/dcds.2019168

Article Contents

2019, Volume 39, Issue 7: 4157-4185. Doi: 10.3934/dcds.2019168

This issue Previous Article Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case Next Article Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $ \mathbb{R}^\text{2n} $

Scaling properties of the Thue–Morse measure

1.
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
2.
Fachbereich Mathematik, Universität Bremen, Postfach 330440, 28359 Bremen, Germany
3.
Research School of Finance, Actuarial Studies and Statistics, Australian National University, 26C Kingsley St, Acton ACT 2601, Australia

Received: September 30, 2018

Published: April 2019

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Abstract

The classic Thue–Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.

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Mathematics Subject Classification: 37D35, 37C45, 52C23.

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Figure 1. The graph of the Birkhoff spectrum $b$ from Eq. (3)

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Figure 2. Illustration of the graphs of $\psi(x)$ (solid line), $\psi(2x)$ (dashed line) and $\psi(4x)$ (dotted line)

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Figure 3. Illustration of $\psi^{}_{3} (x)$ (dotted line) and $\psi^{}_{5} (x)$ (solid line)

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Figure 4. The graph of the pressure function $p$ (solid line) with the two asymptotes $x\mapsto \log(3/2) x$ and $x\mapsto\left(1-x\right)\log(2)$ (dashed lines). The dotted lines are added to illustrate $p(0) = p(1) = \log(2)$

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