Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graphs of continuous but non-affine functions are never self-similar

Carlos Gustavo Moreira, Jinghua Xi, Yiwei Zhang

Abstract

Bandt and Kravchenko \cite{BandtKravchenko2010} proved that if a self-similar set spans $\R^m$, then there is no tangent hyperplane at any point of the set. In particular, this indicates that a smooth planar curve is self-similar if and only if it is a straight line. When restricting curves to graphs of continuous functions, we can show that the graph of a continuous function is self-similar if and only if the graph is a straight line, i.e., the underlying function is affine.

Graphs of continuous but non-affine functions are never self-similar

Abstract

Bandt and Kravchenko \cite{BandtKravchenko2010} proved that if a self-similar set spans , then there is no tangent hyperplane at any point of the set. In particular, this indicates that a smooth planar curve is self-similar if and only if it is a straight line. When restricting curves to graphs of continuous functions, we can show that the graph of a continuous function is self-similar if and only if the graph is a straight line, i.e., the underlying function is affine.

Paper Structure

This paper contains 7 sections, 6 theorems, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Proof of Theorem \ref{['thm:maintheormrigidity']}
  4. Strategy of the proof.
  5. Possible Rotations for Similitudes
  6. Lipschitz Continuity
  7. Proof of Theorem \ref{['thm:maintheormrigidity']}

Key Result

Theorem 1.1

Let $I$ be a compact interval, $f:I\to\mathbb{R}$ be a continuous function and $G=\{(x,f(x)):x\in I\}$. Then the following two statements are equivalent:

Figures (4)

  • Figure 1.1: The graph of Takagi's function
  • Figure 3.1: The iterations of $\mathcal{N},\mathcal{S}$ and $J$ on $S^1$
  • Figure 3.2: $S_i(R)$ frames $f$ on $I_i$
  • Figure 3.3: Illustration of the set $C_2$

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Proposition \ref{['prop:rigidrotation']}
  • Proposition 3.3
  • proof : Proof of Proposition \ref{['prop:Vitalicovering']}
  • Claim 3.4
  • Proposition 3.5
  • ...and 3 more