# Weakly mixing transformations and the Carathéodory definition of measurable sets

Amos Koeller; Rodney Nillsen; Graham Williams

Colloquium Mathematicae (2007)

- Volume: 108, Issue: 2, page 317-328
- ISSN: 0010-1354

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topAmos Koeller, Rodney Nillsen, and Graham Williams. "Weakly mixing transformations and the Carathéodory definition of measurable sets." Colloquium Mathematicae 108.2 (2007): 317-328. <http://eudml.org/doc/286179>.

@article{AmosKoeller2007,

abstract = {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.},

author = {Amos Koeller, Rodney Nillsen, Graham Williams},

journal = {Colloquium Mathematicae},

keywords = {ergodic transformation; mixing; outer measure; Kakutani transformation; measurable set; Carathéodory definition},

language = {eng},

number = {2},

pages = {317-328},

title = {Weakly mixing transformations and the Carathéodory definition of measurable sets},

url = {http://eudml.org/doc/286179},

volume = {108},

year = {2007},

}

TY - JOUR

AU - Amos Koeller

AU - Rodney Nillsen

AU - Graham Williams

TI - Weakly mixing transformations and the Carathéodory definition of measurable sets

JO - Colloquium Mathematicae

PY - 2007

VL - 108

IS - 2

SP - 317

EP - 328

AB - 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.

LA - eng

KW - ergodic transformation; mixing; outer measure; Kakutani transformation; measurable set; Carathéodory definition

UR - http://eudml.org/doc/286179

ER -

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