Weakly mixing transformations and the Carathéodory definition of measurable sets

• Autorzy: Amos Koeller , Rodney Nillsen , Graham Williams
• Języki publikacji: EN

Abstrakty

EN

Let 𝕋 denote the set of complex numbers of modulus 1. Let v ∈ 𝕋, v not a root of unity, and let T: 𝕋 → 𝕋 be the transformation on 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 ψ̃ which is weakly mixing but not strongly mixing. The results here show that the problem of calculating the outer measure of a ψ̃-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 ↦ z² on 𝕋.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2007
• Tom: 108
• Numer: 2
• Strony: 317-328

Daty

wydano

2007

Twórcy

autor

Amos Koeller

• Institut für Mathematik, Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, 14195 Berlin, Germany

autor

Rodney Nillsen

• School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522 Australia

autor

Graham Williams

• School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522 Australia

Identyfikatory

DOI

10.4064/cm108-2-13