Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hölder exponents and fractal structure of level sets of self-affine functions associated with the $Q_s$-representation of numbers

Volodymyr Yelahin, Mykola Moroz

Abstract

We investigate a class of locally complicated self-affine functions defined via the $Q_s$-representation of real numbers. In particular, we compute local Hölder exponents at points with given asymptotic frequencies of digits in their $Q_s$-representation. Furthermore, we establish conditions under which these functions possess continuum level sets. Finally, for self-affine functions satisfying additional conditions, we describe the geometric structure of the set of maximum points and show that this set can be fractal.

Hölder exponents and fractal structure of level sets of self-affine functions associated with the $Q_s$-representation of numbers

Abstract

We investigate a class of locally complicated self-affine functions defined via the -representation of real numbers. In particular, we compute local Hölder exponents at points with given asymptotic frequencies of digits in their -representation. Furthermore, we establish conditions under which these functions possess continuum level sets. Finally, for self-affine functions satisfying additional conditions, we describe the geometric structure of the set of maximum points and show that this set can be fractal.

Paper Structure

This paper contains 6 sections, 28 theorems, 83 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local Hölder Exponents of the Function $f$
  4. Global extrema of the function $f$
  5. Continuum level sets and fractal structure of the set of maximum points of the function $f$
  6. Non-invariance of metric, fractal, and topological characteristics of sets under the transformation $f$.

Key Result

Proposition 2.2

For any positive stochastic vector $Q_s$, every number $x\in[0,1]$ admits at most two $Q_s$-representations. Moreover, if $x$ admits two $Q_s$-representations, then these representations are exactly of the form $\Delta^{Q_s}_{\alpha_1\ldots\alpha_n (0)}$ and $\Delta^{Q_s}_{\alpha_1\ldots\alpha_{n-1}

Figures (5)

  • Figure 1: The graph of the nowhere monotonic singular function $f$ for $q_0=\frac{1}{2}$, $q_1=\frac{3}{10}$, $q_2=\frac{1}{5}$ and $g_0=\frac{1}{5}$, $g_1=\frac{9}{10}$, $g_2=-\frac{1}{10}$.
  • Figure 2: Examples of graphs of the function $f$.
  • Figure 3: Graph of the nowhere monotonic singular function $f$ with continuum level sets from the Example \ref{['ex2']}.
  • Figure 4: Graph of the nowhere differentiable function $f$ with a continuum set of maximum points from the Example \ref{['ex1']}.
  • Figure 5: Geometric construction of the set $C\left[Q_s,V\right]$ from the Example \ref{['ex1']}.

Theorems & Definitions (51)

  • Definition 2.1: AKPrT2011Pr1998
  • Proposition 2.2: AKPrT2011Pr1998, Main theorem on $Q_s$-representation
  • Definition 2.3: AKPrT2011Pr1998
  • Proposition 2.4: AKPrT2011Pr1998
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7: APrT2005Pr1998
  • Proposition 2.8: APrT2005Pr1998
  • Proposition 2.9
  • Proposition 2.10
  • ...and 41 more