How linear can a non-linear hyperbolic IFS be?

TL;DR

This paper constructs and classifies hyperbolic IFSs on [0,1] that exhibit both linear and non-linear behaviors, revealing complex conjugacy and smoothness properties and providing a Livsic-like criterion for conjugacy.

Contribution

It introduces a new class of hyperbolic IFSs with mixed linear and non-linear features and offers a complete classification and conjugacy criteria for these systems.

Findings

Existence of $C^r$-smooth IFSs with linear behavior on the attractor

Classification of these IFSs based on smoothness and conjugacy properties

A Livsic-like condition characterizes when a self-conformal IFS is conjugate to these systems

Abstract

Motivated by a question of M. Hochman, we construct examples of hyperbolic IFSs $\Phi$ 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 $f'\equiv c(\Phi)$ on the attractor and $f''\equiv 0$ for every $f \in \Phi$, yet $\Phi$ 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 $f\in \Phi$. We also show that this condition fails to ensure the existence of a $C^1$-conjugacy in mere $C^1$-regularity.