# An introduction to ergodic theory - Walters P.

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[3] for further discussion and references. Katok [1] connects Liapunov exponents with existence of periodic points.

§10.3 Quasi-invarmnt Measures

In some situations one needs to study a measurable transformation T\ X -*■ X of a probability space (X,bS,m) where T is not measure-preserving but does preserve sets of zero measure (i.e. whenever тф) — 0 then/?i(T_1£' = 0) This occurs when 7 iu a differentiable mip of the interval [0,1] (or of a compact

§10.3 Quasi-invariant Measures

manifold) with a continuous derivative. We then say m is a qmtfl-iM.iriant measure and that Tis a non-singular transformation of(.\ .yj.m). I his raises the problems of studying such transformations and. in particular, of deciding if there is another measure pi on (K,jS) which is equivalent to m (i.e. /1 and m have the name sets of measure aero) und which к an invariant measure for T. There are examples of non-singular transformations having no equivalent invariant probability measure, and examples with no equivalent invariant <7-finite measure. A description of results on these problems is in Friedman's book (Friedman [1]).

Let us consider invertible non-singular transfoimations from now on (i.e. T is bijective and bimeasuruble and 111(B) = 0 llf m(T lB) = 0) Notice that the definition of ergodicity makes sense for non-singular transformations. The class of all ergodic in\ ertible measure-preser ing transformations can be partitioned into four classes as follows. Class I consists of these tranfor-mations T:A -» X where _ t T”x) = 1 for some x e X (i.e. m is con-

centrated on one orbit). If T:(X,.*?,iti) —► (A',.*#, m) is not in class I then it belongs to class II, if there is a probability measure pi on which is

equivalent to m and invariant for T. It belongs to class II, if there is an infinite measure equivalent to m and invai lant for T. Finally T belongs to class III if there is no measure which is equivalent to m and invariant for T. This terminology is used because there is a natural way of associating to tach T a von Neumann algebra which is a factor, and then the above decomposition corresponds to the Murray-von Neumann classification of factors.

A natural notion of isomorphism of non-singular transformations is orbit equivalence. If T,: A, -» Af is an invertible non-singular transformation

i = 1,2, we say T, is orbit equivalent to T2 if there is an invertible nonsingular transformation <j>: A', -» X2 such that for almost all x e А', ф maps the set {Г"х|л t 2} onto the set {T20(x)|/i e Zj.

H. Dye proved that any two transformations of class II, are orbit equivalent and any two transformations of class II, are orbit equivalent W. Krieger introduced the idea of the ratio set of T and this allowed class III to be further divided into classes Шя for/, e [0,1]. He showed that if л e (0,1] then any two members of 111я arc orbit equivalent. For the class III0 the situation is different. These results are descrioed in Sutherland [1].

The above results apply to the non-singular actions of other countable groups on (X,39,m). If G is a topological group then an action of G on (Ais a measurable map ф-.G x A -» X such that ф(е,х) = x Vx e A (where e is the identity element of G) and ф(д1,ф(д2,х)) = ф(д1д2,х)Уд1,д2еС Vx e X. Every action of the integers Z is determined by a bijection T: X -* X by the formula Z x A -» Х:(и,х) -* Tn(x). An action of G is non-singular if for each gsG the transformation x -» ф(д, x) is a non-singular transformation of (A

The results described above hold for actions of a class of countable groups that includes all countable abelian groups. Similar questions can be considered for actions of Lie groups on (X,&#,m) and then one can consider

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10 Appiicat.ons and Other Topics

other types of orbit equivalence; for example we can require that the map ф, in the definition of orbit equivalence, be infinitely differentiable on almost all orbits. Quite a lot is known about orbit equivalence of actions of R (Ornstein [2]).

One can also consider certain questions for countable equivalence relations (see Feldman and Moore [ 1 ] J and also for foliations (see Bowen and Marcus [1], Plante [1]). In particular the ideas of quasi .nvaiiant measure and invariant measure for a foliation has proved useful in the study of differentiable dynamical systems (see references above) and geometry of manifolds (see Thurston’s work on isotopy classes of diffeomorphisms of surfaces).

§10.4 Other Types of Isomorphism

When dealing with measure-preserving transformations which are homeomorphisms of compact metric spaces it makes sense to try to use a more restrictive kind of isomorphism than the usual one. One would say that T, is isomorphic to T2 in the new sense if фТ j = Т2ф for some invertible measure-preserving transformation ф which also ties in with the topological structure in some way. To rtquire that ф be a homeomorphism would be too restrictive. Let us suppose that T, is the two-sided shift on A", = П-®

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