How linear can a non-linear hyperbolic IFS be?

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.

💡 Research Summary

The paper addresses a question raised by Michael Hochman concerning the coexistence of linear and non‑linear behavior in hyperbolic iterated function systems (IFS) on the unit interval. An IFS Φ = {f₀,f₁} is called hyperbolic if each map is a strict contraction with uniformly bounded derivatives and the images are ordered so that the attractor X_Φ is a Cantor set. Hochman’s question asks whether there exist Cʳ‑smooth IFSs (for any 2 ≤ r ≤ ∞) such that on the attractor the first derivative is constant (f′≡c) and the second derivative vanishes (f″≡0) – i.e. the system is “linear” in the sense of having no curvature – yet the IFS is not Cʳ‑conjugate to any self‑similar (affine) system.

The authors introduce the notion of a pseudo‑affine IFS: a C¹‑system for which there exists a slope λ∈(0,½) with f′(x)=λ for every x∈X_Φ and every f∈Φ. When the maps are C² or smoother, pseudo‑affinity is equivalent to the linearity condition f″≡0 on X_Φ. The main result (Theorem 2) shows that for every λ∈(0,½) and every regularity level s∈