By A.A. Tempelman

This quantity is dedicated to generalizations of the classical Birkhoff and von Neuman ergodic theorems to semigroup representations in Banach areas, semigroup activities in degree areas, homogeneous random fields and random measures on homogeneous areas. The ergodicity, blending and quasimixing of semigroup activities and homogeneous random fields are regarded as good. particularly homogeneous areas, on which all homogeneous random fields are quasimixing are brought and studied (the *n*-dimensional Euclidean and Lobachevsky areas with *n*>=2, and all easy Lie teams with finite centre are examples of such areas. additionally handled are purposes of basic ergodic theorems for the development of particular informational and thermodynamical features of homogeneous random fields on amenable teams and for proving normal types of the McMillan, Breiman and Lee-Yang theorems. A variational precept which characterizes the Gibbsian homogeneous random fields when it comes to the explicit loose power can also be proved. The booklet has 8 chapters, a few appendices and a considerable record of references.

For researchers whose works comprises likelihood conception, ergodic idea, harmonic research, degree conception and statistical Physics.

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**Sample text**

We can extend the operators and S: (from LOI n LP n L2 into L2) to continuous operators 8; in L2. 12), 118~1I < 1. Clearly, 8 2 : x 1-+ 8~ is a representation of X in L2 which is consistent with SOl and 8 P and, since X is a group, this representation is unitary. 16. 7. Averageability with respect to a-contractive representations in Ll. Let (n, :F, m) be a measure space. 5. Let 1 < a <00. A representation S in L~ will be called a-contractive if it is a contractive representation and is consistent with a contractive representation 8(01) in L~.

0, or m(f) -::j:. 0, respectively. We shall consider some simple properties of T- and I-means and of 1'- and I-quasi-averageable functions; for brevity we shall formulate statements only for "right" objects; formulations for the "left" ones can be obtained simply by symmetry. 1. ) when B = C. Proof. Statement (i) follows from the property (ii) of the operators RA II : M(r)(f) IIB= n->oo lim II RAnf IlwB ~ IlfllwB if An E As(r)(f, M(r)(f)). Statements (ii) and (iii) can be proved similarly. 2. If f is a B-valued r-quasi-averageable function on X and I is a continuotts linear mapping of B into a Banach space D, then the D-valued function 1 0 f is also r-qtwsi-averageable and l(9Jt(r\f)) C 9Jt(r)(l 0 f).

Let f E tJJB, a E m(r)(f), b E m(l)(f), {An} E As(r)(f,a) and {v n } E AS(i)(f, b). 4, {v n * An} E E As(r)(f, a) n AS(l)(f, b), and, consequently, a = b = M(f). o We conclude this section by a discussion of the· problem of "invariance" of the sets m(r)(f) and m(l)(f) with respect to the action of the operators Rv and Lv on the function f (v E 15). 6. For any f E c m(l)(fL v ) c m(r)(Rvf) {fj B, c m(r)(fL v ), m(l)(f) c m(l)(Rvf). m(r)(f) Proof. Let a E m(r)(f) and {An} E As(r)(f, a). 1,) Chapter 1 14 Thus a E mer) (f L,,).