English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

§ 1. Let T be an ergodic measure preserving transformation of a Lebesgue measure space (Q, 23, P), P(O)=1, that is, Tis a one to one mapping from Q onto itself, bimeasurable (T9J = 93), measure preserving (P(T"1A) = P(A) for A in S3) and ergodic (every measurable function /(«) with f(Tco)=f((o) a.e. is constant a.e.). For measure preserving transformations 17 and 17' we write 17 = 17' if P(UcQ^U'o)) = Q and U^U' otherwise. A measure preserving transformation 17 of (O5 S, P) is called a p-ih root of TQ?^2) if l/* = T. A 1-parameter group {If,} of measure preserving transformations of (Q, S3, P) (i.e. 17,+s= 17,17S for — oo < f, s < -f oo) is called a measurable flow if (co, i)-»Uta) is a measurable mapping from £2 x R onto £2. If there exists a measurable flow {Ut} with 17! = T, Tis said to be embeddable in a measurable flow. The existence of a p-th root or an embedding measurable flow has been one of problems in ergodic theory. It is obvious that the existence of an embedding measurable flow of T implies the existence of a p-th root of Tfor every p^2 and also that a p-ih root of T(if exists) and an embedding measurable flow of T(if exists) are ergodic. A measure preserving transformation 17 of (Q, 23, P) is said to commute with T if UT= TU. We denote by C(T) the group consisting of all measure preserving transformations each of which commutes with T and call it the centralizer of T. Since a p-th root of T(if exists) and a transformation Ut for fixed t in an embedding measurable flow {Ut} of T (if exists) are in C(T), we may expect to solve the existence problem of roots and an embedding measurable

Centralizer of an ergodic measure preserving transformation