Linear dynamics of an operator associated to the Collatz map

Béhani, Vincent

Type de document :

Article dans une revue scientifique: Article original

DOI :

10.1090/proc/16627

URL permanente :

http://hdl.handle.net/20.500.12210/121446

Titre :

Linear dynamics of an operator associated to the Collatz map

Auteur(s) :

Béhani, Vincent [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Université de Lille

Titre de la revue :

Proceedings of the American Mathematical Society

Éditeur :

American Mathematical Society

Date de publication :

2024-01-11

ISSN :

0002-9939

Mot(s)-clé(s) en anglais :

Frequent Hypercyclicity
Ergodicity
Collatz conjecture
Hypercyclicity
Chaotic operators

Discipline(s) HAL :

Mathématiques [math]

Résumé en anglais : [en]

In this paper, we study the dynamics of an operator T \mathcal T naturally associated to the so-called Collatz map , which maps an integer n ≥ 0 n \geq 0 to n / 2 n / 2 if n n is even and 3 n + 1 3n + 1 if n n is odd. This ...
Lire la suite >In this paper, we study the dynamics of an operator T \mathcal T naturally associated to the so-called Collatz map , which maps an integer n ≥ 0 n \geq 0 to n / 2 n / 2 if n n is even and 3 n + 1 3n + 1 if n n is odd. This operator T \mathcal T is defined on certain weighted Bergman spaces B ω 2 \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that T \mathcal T is hypercyclic on B ω 2 \mathcal B ^2 _\omega , independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight ω \omega , we show that T \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic.Lire moins >

Langue :

Anglais

Comité de lecture :

Oui

Audience :

Internationale

Vulgarisation :

Non

Projet ANR :

Frontières de la théorie des opérateurs
Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

Date de dépôt :

2025-01-24T13:56:14Z

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