By E. M. Friedlander, M. R. Stein

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**Example text**

The inequality ~ introduces the partial order in the set of all partitions. The maximal element of this set is the partition G of M into separate points, while the minimal one is the partition v whose unique element is M itself. For any family of measurable partitions ga} the partitions SUPa ~a (notation Va ~a) and infa ~a (notation I\a ~a) are well defined and are called the measurable product and measurable intersection of {~a} respectively. 4t whose elements are the subsets of M consisting (mod 0) of the entire elements of ~.

Consider the integer valued function kE on E such that kE(x) = min{n ~ 1: T"x E E}. It follows from the Poincare recurrence theorem that kE is well defined for almost all X E E. The function kE is known as the return time function into E. For T ergodic the following Kac formula is true: Chapter 1. l(E)r 1. l(E)r 1. For almost every x E E define the transformation TE by TEx = TkE(X)X, X E E. A, AcE. 1. The automorphism TE is called"the induced automorphism constructed from the automorphism T and the set E.

Sinai Ml(l = 1,2), then for any n ~ 1 we set d,. ((TI , ~ I ), (Tz , ~ z )) = d( {Tlk ~ 1 } Z-I , { Tzk~ 2 } Z-1 ). 3. 1 ) The right-hand side of the latter equality may be rewritten with limn instead of upper limit, since such limit always exists. This remark shows that the triangle inequality hold for II. ff = (io, ... , i n- l ), (ik are integers), given by n C~l) n··· n C~l) V(l)(§) = j1 I (C~l) n lO,O ll,1 In __ ) l,n-l' where ql:k" .. , C~~~ are the elements of the partition ~lk), k = 0, 1, ...