Linear dynamics of an operator associated to the Collatz map

Béhani, Vincent

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1090/proc/16627

Title :

Linear dynamics of an operator associated to the Collatz map

Author(s) :

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

Journal title :

Proceedings of the American Mathematical Society

Publisher :

American Mathematical Society

Publication date :

2024-01-11

ISSN :

0002-9939

English keyword(s) :

Frequent Hypercyclicity
Ergodicity
Collatz conjecture
Hypercyclicity
Chaotic operators

HAL domain(s) :

Mathématiques [math]

English abstract : [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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

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

Collections :