By Janet Whalen Kammeyer
This monograph bargains a vast investigative software in ergodic conception and measurable dynamics. the incentive for this paintings is that one may possibly degree how related dynamical structures are by means of asking how a lot the time constitution of orbits of 1 procedure has to be distorted for it to develop into the opposite. various regulations at the allowed distortion will result in diversified limited orbit equivalence theories. those comprise Ornstein's Isomorphism conception, Kakutani Equivalence thought and a listing of others. by way of placing such regulations in an axiomatic framework, a basic technique is constructed that encompasses all of those examples at the same time and offers perception into tips on how to search additional functions.
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Additional info for Restricted Orbit Equivalence for Actions of Discrete Amenable Groups
Sample text
Heicklen's conclusion is that two actions a ~ /J iff there is a ip G F so that Tai' and T'1 have identical Ht orbits for all i. As Ta and T0"" are conjugate (by \p of course) T01 and T^ are f related and if two actions are f related they can be realized as two such actions. The family of sizes exhibits two very interesting properties. The first is due to Vershik who proved a lacunary isomorphism theorem for such groups [58]: for any two actions U and V, if the /-,- are chosen to grow rapidly enough, then the two actions are f related.
The full-group is separable in the L1-topology and so, by Axiom 2, is separable in the ma-topology and we can find a countable collection dense in all the mx or m^-topologies. Hence It remains to see that the sets &(4>,\p,£) are open in fx. Suppose (4>i)a e &(4>,ip,s) and hence there is an e > 0 with , \p) < ma(((/),)o,0, ((t>i)a\p) + e + e. ' e G(
Thus, since Q is a contraction, for (f> G F, mp(Q(P(cl>)),cl>) = 0. Hence for all <> / G {tp,mp), we see that Thus, for a,b £ (Yp,mp), mp(a,b) = mll{Q(P(a)),Q(P(b)))