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The multifractal nature of a parametrized family of von Koch functions

Zoltán Buczolich, Yann Demichel, Stéphane Seuret

TL;DR

The paper studies the generalized von Koch function $F^ olinebreak[1]^ olinebreak[ lambda}$, constructed by iterating a geometric rule and parameterized by $ olinebreak[1] olinebreak[1]$, in the range $ olinebreak ( frac{1}{6}, frac{5}{6})$, proving that $F^ olinebreak[1]^ olinebreak[1]$ is continuous everywhere but nowhere differentiable and possesses a rich multifractal structure. The analysis couples a dynamical system on $[0,1]$ with auxiliary self-similar measures $ olinebreak[1]{ abla}$ and tailored iterated function systems to connect the pointwise regularity of $F^ olinebreak[1]$ to the local dimension of $ olinebreak[1]{ abla}$ via the Lyapunov-type slopes $m_n(x)$. A key finding is that the multifractal spectrum $d_{F^ olinebreak[1]}( olinebreak abla)$ is a shifted and truncated version of the spectrum of $ olinebreak[1]{ abla}$, with a phase transition at $ olinebreak[1]={1}/3$, and explicit endpoints $ olinebreak ilde{ olinebreak { olinebreak olinebreak }}_{ olinebreak[1], ext{min}}=1- frac{ olinebreak ext{log}(6 ename{  }+1)}{ ename{ ext{log}} 6}$ and $ olinebreak ilde{ olinebreak { olinebreak  } }_{ olinebreak[1], ext{L}}= 1- frac{ ext{log}(36 olinebreak[1] olinebreak ext{  }^2-1)}{4 ename{ ext{log}} 3+2 ename{ ext{log}} 6}$. The work also reveals a non-self-similarity of $F^ olinebreak[1]$ and details the spectrum’s behavior across $ olinebreak [ frac{2}{6}, frac{1}{3}]$ and $ olinebreak ( frac{1}{3}, frac{5}{6})$, including the role of $ olinebreak $ as a phase transition point and the interplay between $d_{F^ olinebreak[1]}$ and the multifractal spectrum $ au^*_{ olinebreak[1]}$ of $ olinebreak[1]{ abla}$.

Abstract

In a famous paper published in 1904, Helge von Koch introduced the curve that still serves nowadays as an iconic representation of fractal shapes. In fact, von Koch's main goal was the construction of a continuous but nowhere differentiable function, very similar to the snowflake, using elementary geometric procedures, and not analytical formulae. We prove that a parametrized family of functions (including and) generalizing von Koch's example enjoys a rich multifractal behavior, thus enriching the class of historical mathematical objects having surprising regularity properties. The analysis relies on the study of the orbits of an underlying dynamical system and on the introduction of self-similar measures and non-trivial iterated functions systems adapted to the problem.

The multifractal nature of a parametrized family of von Koch functions

TL;DR

The paper studies the generalized von Koch function , constructed by iterating a geometric rule and parameterized by , in the range , proving that is continuous everywhere but nowhere differentiable and possesses a rich multifractal structure. The analysis couples a dynamical system on with auxiliary self-similar measures and tailored iterated function systems to connect the pointwise regularity of to the local dimension of via the Lyapunov-type slopes . A key finding is that the multifractal spectrum is a shifted and truncated version of the spectrum of , with a phase transition at , and explicit endpoints and . The work also reveals a non-self-similarity of and details the spectrum’s behavior across and , including the role of as a phase transition point and the interplay between and the multifractal spectrum of .

Abstract

In a famous paper published in 1904, Helge von Koch introduced the curve that still serves nowadays as an iconic representation of fractal shapes. In fact, von Koch's main goal was the construction of a continuous but nowhere differentiable function, very similar to the snowflake, using elementary geometric procedures, and not analytical formulae. We prove that a parametrized family of functions (including and) generalizing von Koch's example enjoys a rich multifractal behavior, thus enriching the class of historical mathematical objects having surprising regularity properties. The analysis relies on the study of the orbits of an underlying dynamical system and on the introduction of self-similar measures and non-trivial iterated functions systems adapted to the problem.

Paper Structure

This paper contains 18 sections, 33 theorems, 181 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The dynamical system and the associated decomposition of $x\in[0,1]$
  3. Estimates on the pointwise regularity of ${F^\lambda}$ when $\lambda\in(\frac{1}{6},\frac{5}{6})$
  4. The approximating functions ${F_n^{\lambda}}$
  5. Proof of the convergence part of Theorem \ref{['theo-finite']} and non-differentiability of ${F^\lambda}$.
  6. The increments of ${F^\lambda}$
  7. Estimates on the pointwise Hölder exponents of ${F^\lambda}$
  8. Proof of Theorem \ref{['*thselfs']}: non-self-similarity of ${F^\lambda}$
  9. The measure ${\mu_\lambda}$ and its multifractal spectrum
  10. Recalls on the multifractal properties of ${\mu_\lambda}$
  11. Behavior of $d_{\mu_\lambda}$ in terms of $\lambda$
  12. The lower bound for $d_{F^\lambda}$ when $\lambda\in(\frac{1}{6},\frac{5}{6})$ and the decreasing part of $d_{F^\lambda}$
  13. The upper bound for the increasing part of $d_{F^\lambda}$ when $\lambda\in [\frac{\sqrt2}{6},\frac{1}{3})$
  14. Construction of the IFS
  15. The family of the intervals ${\mathbf J}_j'$
  16. ...and 3 more sections

Key Result

Theorem 1.1

When $\lambda\in (\frac{1}{6},\frac{5}{6})$, the sequence $({F_n^{\lambda}})_{n\geq1}$ converges uniformly to ${F^\lambda}$, which is continuous but nowhere differentiable. In addition, the set $\cal{V}$ of those points $x$ where the derivative of ${F^\lambda}$ is infinite has Hausdorff dimension $s When $\lambda \geq \frac{5}{6}$, there are points $x\in[0,1]$ at which ${F^\lambda}(x)=\lim_{n\to\i

Figures (10)

  • Figure 1: The basic geometric operation $\Omega^\lambda$ transforms a given line segment $AB$ into the polygonal line $ACDEB$ made up of $4$ consecutive line segments.
  • Figure 2: The generalized von Koch function ${F^\lambda}$ for various parameters $\lambda$. The original function by von Koch corresponding to the choice $\lambda=\frac{\sqrt3}{6}$ is shown by the thick red curve.
  • Figure 3: When $\lambda\in(0,\frac{1}{6})$, the operation $\Omega^\lambda$ may lead to new slopes all with the same sign, hence a monotonous part in the graph of ${F_n^{\lambda}}$.
  • Figure 4: Illustration for the dynamics of $T$ associated with the construction of the function ${F^\lambda}$.
  • Figure 5: The multifractal spectrum $d_{F^\lambda}$ of ${F^\lambda}$ for different values of $\lambda\in(\frac{\sqrt2}{6},\frac{5}{6})$. The spectrum of the original von Koch function corresponding to the choice $\lambda=\frac{\sqrt3}{6}$ is shown by the thick red curve.
  • ...and 5 more figures

Theorems & Definitions (77)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 67 more