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How linear can a non-linear hyperbolic IFS be?

Amir Algom, Snir Ben Ovadia, Federico Rodriguez Hertz, Mario Shannon

Abstract

Motivated by a question of M. Hochman, we construct examples of hyperbolic IFSs $Φ$ on $[0,1]$ where linear and non-linear behaviour coexist. Namely, for every $2\leq r \leq \infty$ we exhibit the existence of a $C^r$-smooth IFS such that $f'\equiv c(Φ)$ on the attractor and $f''\equiv 0$ for every $f \in Φ$, yet $Φ$ is not $C^t$-smooth for any $t>r$, nor $C^r$-conjugate to self-similar. We provide a complete classification of these systems. Furthermore, when $r>1$, we give a necessary and sufficient Livsic-like matching condition for a self-conformal $C^r$-smooth IFS to be conjugated to one of these systems having $f''=0$ on the attractor, for every $f\in Φ$. We also show that this condition fails to ensure the existence of a $C^1$-conjugacy in mere $C^1$-regularity.

How linear can a non-linear hyperbolic IFS be?

Abstract

Motivated by a question of M. Hochman, we construct examples of hyperbolic IFSs on where linear and non-linear behaviour coexist. Namely, for every we exhibit the existence of a -smooth IFS such that on the attractor and for every , yet is not -smooth for any , nor -conjugate to self-similar. We provide a complete classification of these systems. Furthermore, when , we give a necessary and sufficient Livsic-like matching condition for a self-conformal -smooth IFS to be conjugated to one of these systems having on the attractor, for every . We also show that this condition fails to ensure the existence of a -conjugacy in mere -regularity.

Paper Structure

This paper contains 24 sections, 15 theorems, 119 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries
  4. Regularity
  5. Binary Cantor Sets and Dynamical Proportions
  6. Hyperbolic structures on Cantor sets
  7. Some properties of self-conformal IFSs
  8. pseudo-Affine hyperbolic IFS.
  9. Asymptotically constant dynamical proportions
  10. The key construction
  11. $C^s$-conjugacy classes of pseudo-affine hyperbolic IFSs
  12. On the proof of Theorem \ref{['Theorem A']}
  13. Case (a): $C^s$-conjugated to self-similar
  14. Case $s=1$.
  15. Case $1<s<\infty$.
  16. ...and 9 more sections

Key Result

Theorem 2

For every parameter $0<\lambda<1/2$ and $1\leq s\leq \infty$, there exist hyperbolic pseudo-affine IFSs $\Phi \subseteq C^s([0,1])$ with slope $\lambda$, satisfying that $\Phi \not \subseteq C^t([0,1])$ for every $s<t$. In the case $s=\infty$, this means that there exist such IFSs that are $C^\infty

Figures (3)

  • Figure 1: On the left: Labeling of cylinders and gaps. On the right: The action of the IFS.
  • Figure 2: The proportions $\lambda_i:\mathcal{W}\to(0,1)$, $i=0,1$ define a unique Cantor set $X$ with a marking $\{I_w\}_{w\in\mathcal{W}}$ of its gaps such that $|I_{iw}|/|I_w|=\lambda_i(w)$. We see here how to define an IFS $\Phi=\left\lbrace f_0,f_1 \right\rbrace$ that preserves this cantor set.
  • Figure 3: Edges in blue correspond to $\theta_i(w)=0$, otherwise colored in red.

Theorems & Definitions (44)

  • Theorem 2
  • Corollary 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • Lemma 8
  • proof
  • Theorem 9: Classification of hyperbolic $C^{1+\alpha}$-structures, Sullivan1987ratio-BedfordFisher1997Ratio.
  • Theorem 10: Rigidity, Sullivan1987ratio-BedfordFisher1997Ratio
  • Definition 11
  • ...and 34 more