How linear can a non-linear hyperbolic IFS be?

2024/10/29 by Amir Algom, Snir Ben Ovadia, Algom, Amir +5 · 1 citation
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Mathematical Dynamics and Fractals #Cellular Automata and Applications

paper · pdf · doi:10.48550/arxiv.2410.22145

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≤ r ≤ ∞ we exhibit the existence of a C^r-smooth IFS such that f'≡ c(Φ) on the attractor and f''≡ 0 for every f ∈ 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∈ Φ. We also show that this condition fails to ensure the existence of a C¹-conjugacy in mere C¹-regularity.

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