By L. Auslander, R. Tolimieri

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Extra resources for Abelian Harmonic Analysis, Theta Functions and Functional Analysis on a Nilmanifold

Example text

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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