By Greg Hjorth

Activities of Polish teams are ubiquitous in arithmetic. In definite branches of ergodic concept and useful research, one unearths a scientific examine of the gang of measure-preserving adjustments and the unitary workforce. In common sense, the research of countable types intertwines with effects about the activities of the limitless symmetric workforce. this article develops the speculation of Polish team activities completely from scratch, eventually offering a coherent thought of the ensuing orbit equivalence periods that can permit whole category through invariants of an indicated shape. The publication concludes with a criterion for an orbit equivalence relation classifiable by way of countable constructions thought of as much as isomorphism. This self-contained quantity deals an entire therapy of this energetic sector of present examine and develops a tough basic idea classifying a category of mathematical gadgets as much as a few appropriate suggestion of isomorphism or equivalence. Greg Hjorth bought the Carol Karp Prize for striking paintings on turbulence and countable Borel equivalence kin from the organization of Symbolic good judgment.

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3: (x)dl Since A was arbitrary, that an i n t e g r a b l e measure the e r g o d i c equation(ii) functions ~ theorem which satisfyis ab- to an i n d i c a t o r CA(Tkx) = E(CA II J} = Z(A) by the e r g o d i c i t y Let ~o be an i n t e g r a b l e ~9(x) p(V(k)x)A(k) of an i n v a r i a n t to I. A p p l y i n g n-i 1 lim [ Z n÷~ k=O is Z~(A) shows is a d e n s i t y continuous which theorem. reasoning solutely ! I = k B(b I .... ,bn_l)nA by the c h a n g e - o f - v a r i a b l e s = Z ~-i k for x e B ( b l , .

K 9 I ..... kg)x) ~ ( k I ..... k ) (x) - A(k I .... k ) ( y ) ] + + A(k I ..... k ) (y) Using lemma ~ o ( V ( k l , .... 5 and l~o(V(kl, .... k)y) <_ K d ( V ( k l , . . 7: as in lemma - ~o~V(kl ..... k~)x) I ! )y , V(kl, .... k )x) <_ K2c(~) are b o t h contained Let C O as in lemma Proof: b = b(m,M,K) From since V(k I ..... k )y in B ( k l , . . 6 and the s e q u e n c e (C~)ve N d e f i n e d lemma - ap I < ba(9) and a = / C ° are constant. 6 O < go ~ Go with got9 ! C~+~ ! GOC ~ uniformly in ~ and U.

K s ) and - 34 - log d I + log p(B(k I ..... ks)) <_ log A(k I ..... 4 resp. 7 w i t h constants dl,d 2 > O we see lim ~1 log ~(B(k I .... e. e. 4 and § 8. 6 using A O(s+n+l) >A °(s+j) lemma 1 < j < n. Since i I p is e q u i v a l e n t with respect integrable to I, it is in p a r t i c u l a r to I. e. continuous the e x i s t e n c e of an with d 1 is known. One can easily verify the of Gauss 1 1 = log 2 " l+x p(x) by a direct calculation. The shape of the d e n s i t y function is u n k n o w n for n > 2.

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