How linear can a non-linear hyperbolic IFS be?

Abstract

Motivated by a question of M. Hochman, we construct examples of hyperbolic IFSs $\hat{I}¦$ on $[0,1]$ where linear and non-linear behaviour coexist. Namely, for every $2\le r\le \infty$ we exhibit the existence of a $C^r$-smooth IFS such that $f^{\prime }\equiv c(\hat{I}¦)$ on the attractor and $f^{\prime \prime }\equiv 0$ for every $f\in \hat{I}¦$, yet $\hat{I}¦$ 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 $f^{\prime \prime }=0$ on the attractor, for every $f\in \hat{I}¦$. We also show that this condition fails to ensure the existence of a $C^1$-conjugacy in mere $C^1$-regularity.