Weakly mixing transformations and the Carathéodory definition of measurable sets

Amos Koeller¹ ; Rodney Nillsen² ; Graham Williams²

¹ Institut für Mathematik Fachbereich Mathematik und Informatik Freie Universität Berlin Arnimallee 2-6 14195 Berlin, Germany
² School of Mathematics and Applied Statistics University of Wollongong Wollongong, NSW 2522 Australia

Colloquium Mathematicum, Tome 108 (2007) no. 2, pp. 317-328.

Résumé

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 ${\Bbb T}$.

Détail

DOI : 10.4064/cm108-2-13

Keywords: bbb denote set complex numbers modulus bbb root unity bbb rightarrow bbb transformation bbb given known problem calculating outer measure t invariant set leads condition which formally has close resemblance carath odorys definition measurable set ergodic theory terms weakly mixing there example due kakutani transformation widetilde psi which weakly mixing strongly mixing results here problem calculating outer measure widetilde psi invariant set leads condition formally resembling carath odory definition transformation methods bring out detailed behaviour kakutani transformation above mentioned results kakutani transformation apply strongly mixing transformation mapsto nbsp bbb