# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## Weakly mixing transformations and the Carathéodory definition of measurable sets

### Tom 108 / 2007

Colloquium Mathematicum 108 (2007), 317-328 MSC: Primary 37A05; Secondary 28A05. DOI: 10.4064/cm108-2-13

#### Streszczenie

Let ${\Bbb T}$ denote the set of complex numbers of modulus $1$. Let $v\in{\Bbb T}$, $v$ not a root of unity, and let $T:{\Bbb T}\rightarrow {\Bbb T}$ be the transformation on ${\Bbb T}$ given by $T(z)=vz$. It is known that the problem of calculating the outer measure of a $T$-invariant set leads to a condition which formally has a close resemblance to Carathéodory's definition of a measurable set. In ergodic theory terms, $T$ is not weakly mixing. Now there is an example, due to Kakutani, of a transformation $\widetilde \psi$ which is weakly mixing but not strongly mixing. The results here show that the problem of calculating the outer measure of a $\widetilde \psi$-invariant set leads to a condition formally resembling the Carathéodory definition, as in the case of the transformation $T$. The methods used bring out some of the more detailed behaviour of the Kakutani transformation. The above mentioned results for $T$ and the Kakutani transformation do not apply for the strongly mixing transformation $z\mapsto z^2$ on ${\Bbb T}$.

#### Autorzy

• Amos KoellerInstitut für Mathematik
Fachbereich Mathematik und Informatik
Freie Universität Berlin
Arnimallee 2-6
14195 Berlin, Germany
e-mail
• Rodney NillsenSchool of Mathematics and Applied Statistics
University of Wollongong
Wollongong, NSW 2522 Australia
e-mail
• Graham WilliamsSchool of Mathematics and Applied Statistics
University of Wollongong
Wollongong, NSW 2522 Australia
e-mail

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