How linear can a non-linear hyperbolic IFS be?
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914413848559616 |
|---|---|
| author | Algom, Amir Ovadia, Snir Ben Hertz, Federico Rodriguez Shannon, Mario |
| author_facet | Algom, Amir Ovadia, Snir Ben Hertz, Federico Rodriguez Shannon, Mario |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22145 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | How linear can a non-linear hyperbolic IFS be? Algom, Amir Ovadia, Snir Ben Hertz, Federico Rodriguez Shannon, Mario Dynamical Systems 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. |
| title | How linear can a non-linear hyperbolic IFS be? |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.22145 |