# Ergodic Theory of Random Transformations by Yuri Kifer

By Yuri Kifer

Ergodic conception of dynamical structures i.e., the qualitative research of iterations of a unmarried transformation is these days a good constructed conception. In 1945 S. Ulam and J. von Neumann of their brief be aware [44] prompt to review ergodic theorems for the extra basic scenario whilst one applies in flip assorted transforma tions selected at random. Their software was once fulfilled through S. Kakutani [23] in 1951. 'Both papers thought of the case of differences with a standard invariant degree. lately Ohno [38] spotted that this situation used to be over the top. Ergodic theorems are only the start of ergodic concept. between additional significant advancements are the notions of entropy and attribute exponents. the aim of this booklet is the examine of the diversity of ergodic theoretical houses of evolution techniques generated by means of self reliant purposes of alterations selected at random from a definite classification in response to a few likelihood distribution. The booklet shows the 1st systematic remedy of ergodic thought of random modifications i.e., an research of composed activities of self reliant random maps. This manage permits a unified method of many difficulties of dynamical platforms, items of random matrices and stochastic flows generated by way of stochastic differential equations.

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S. J) < 00. s. J) = (y EM '\ G: where d(x,G) = yEG inf d(x,y). L8.. 2. 11) Then there exists a non-random Limit Ao(G) = lim llog n .... s. 13) Proof. 11) says that the sequence log A~(G,w), n = 1,2, ... 12). s. Clearly, ~(G ,CJ) decreases when 0+ 0 and so does i\/j(G). 3. 11), as welL The number i\(G) is connected with the stability properties of an f-invariant set G. s. for all n = 1,2, ... pro~ O. s. as n ~ 00 then we shall call G asymptotically stable. 1. If i\( G) < 0 and the f-invariant set G is stable then it is asymptotically stable.

H~(rp). A and write These numbers are called the entropy of rp with and the entropy of rp, respectively. Next we shall discuss main properties of the entropy. 3. 6). (rp,T/). (rp,T/). ,,) + H~(~I:)T/)). (rp,T/). o ",-in Proof. 2 (iii) the assertion follows. (\/ rp---i~I:A:\/ rp~TJ)v--4). 8) yields the assertion (iv). 3 the following important property of the entropy. 4. Suppose that rp : M ..... M preserves JJ. 6). then h/(rpk) = kh/(rp) for any integer k > O. 9} Proof. 10) Indeed, lim n ... 11) where each time the supremum is taken over finite partitions ~ and 7].

Ir1~) ..... s. Proof. 1. Put -52- n+m-1 a n +m ~ ~ + Hp(nf-1(~v \ / (fio ... ofn+1)-1~)). m . Set n+m-l ~n m = ~ v \ / (fi o ... ofn+l)-l~ then . m =-J ... J(J ... J ~ dp ~(p((fno ... '" dm(f n+1) ... J ~ ~(J ... Jp(f nO· .. of l)-lA )dm(f 1) ... '" dm(f n+l) ... 1. o ir-1~)dp= J amdp. e. is subadditive. 22) exists. 2(x). -53n+m-J Cn,m = Hp(~v . \ / (ft o ... ofn+l)-l~) = am 0 ~=n+l 'd n and so a n +m ~ an + am 'd n . 4. If p E P(M) is P-invariant then the number hp(f) = sup hp(f,~), where the supremum is taken over all finite partitions of M is called the entropy of a random transformation f, having the distribution tn, with respect to the measure p.