By Edwin Beckenbach

College of CaliforniaEngineering and actual sciences extension series.Includes bibliographies.Based at the Statewide lecture sequence on combinatorial arithmetic provided through the collage of California, collage Extension, Engineering and actual Sciences department, in 1962.

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**Example 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)))