JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Artykuły Online First

## Collatz map as a non-singular transformation

Studia Mathematica MSC: Primary 11B75; Secondary 37A40, 37A44 DOI: 10.4064/sm230913-14-12 Opublikowany online: 11 April 2024

#### Streszczenie

Let $T$ be the map defined on $\mathbb{N} =\{1,2,\ldots \}$ by $T(n) = {n}/{2}$ if $n$ is even and $T(n) = (3n+1)/2$ if $n$ is odd. Consider the dynamical system $(\mathbb {N}, 2^{\mathbb {N}}, T,\mu )$ where $\mu$ is the counting measure. This dynamical system has the following properties:
1. There exists an invariant finite measure $\gamma$ such that $\gamma (A) \leq \mu (A)$ for all $A \subset \mathbb{N}$.
2. For each function $f\in L^1(\mu )$ the averages $\frac {1}{N} \sum _{n=1}^N f(T^nx)$ converge for every $x\in \mathbb{N}$ to $f^*(x)$ where $f^* \in L^1(\mu )$.
We also show that the Collatz conjecture is equivalent to the existence of a finite measure $\nu$ on $(\mathbb{N} , 2^{\mathbb{N} })$ making the operator $Vf = f\circ T$ power bounded in $L^1(\nu )$ with conservative part $\{1,2\}$.

#### Autorzy

• I. AssaniDepartment of Mathematics
UNC Chapel Hill
Chapel Hill, NC 27599, USA
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek