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# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 2 3 4 5 6 < 7 > 8 9 10 11 12 13 .. 99 >> Next (6) If –ď is a subgroup of G then H = {g e G|y(g) = 1 Vy e –ď} is a closed subgroup of G and (–Ī/fl) = –ď. Notice it makes sense for elements of –ď to act on G/II and this result says that these are the only continuous homomor-phisms of G/H into K.
(7) If H is a closed subgroup of G and H –§ G there exists –į —É e 6, —É —Ą 1 such that y(h) = 1 Vlie H. (We shall write this —É(II) = 1.)
(8) Let G be compact. The members of <3 arc mutually orthogonal members of L2(m), where in is Haar measure.
Proof. It suffices to show
jc y(x)dm(x) = 0 ify^l.
If a e G, then, since m is a Haar measure,
jy(x) dm(x) = ^y(ax) dm(x) = y(a) JvM dm(x).
Choosing –ł so that y(a) –§ 1 we have ¬ßy(x)dm(x) = 0. ‚Ė°
(9) If G is compact, the members of G form an orthonormal basis for L2(m) where m is Haar measure.
This is part of the Peter-Weyl theorem and can be easily deduced from the Stone-Weierstrass theorem, which implies that finite linear combinations of characters are dense in C(G), the space of complex-valued continuous functions of G.
Therefore for each / e L2(m) there are uniquely determined complex numbers ay such that / = Y.-,ec ail *n L2(m). This means Ve > 0 there is a finite subset J0 of G such that if J is finite and J0cz J thenjj/ ‚ÄĒ ayy ]2 < –≥.. Only a countable number of ay are non-zero. We have ¬£./ec |–į—É|‚Äú = ||/ \\ < cc. This representation of / is called the Fourier series of /. When G = –ö = (te –°11;| = 1} the Fourier series of / is the classical Fourier series /(z) = ^ a‚Äěz" since G consists of the maps y(z) = z",ne Z.
(10) If A :G -*‚Ė† G is an endomorphism we can define the dual endomorphism A\G -> G by Ay ‚ÄĒ —É ¬į A, —É e G. It is easy to see that A is one-to-one if and only if A is onto, and A is onto if and only if A is one-to-one. Therefore A is an automorphism if and only if A is an automorphism.
Recall that for compact groups G, G is metrisable iff G has a countable topological base.
—č
–ě Preliminaries
¬ß0.8 Endomorphisms of Tori
We shall view llic //-torus in lwo ways- multiplicatively as K" and additively as R"'Z" whore R" is the additive group of ¬ę-dimensional Euclide#n spacc and Z" is the subgroup of R" consisting of the points with integer coordinates. A topological group isomorphism from K" to R"/Z" is given by
(tr'"'1,... ,c2nix‚ÄĚ) –ł-* (x,,. .. ,x‚Äě) + Z".
Theorem 0.14
(i) Ererv closed subgroup of –ö is either –ö nr is a finite cyclic group consisting of all p-tli roots of unit\ for some integer p > 0.
(ii) The only automorphisms of –ö are the identity and the map zi‚ÄĒ>r-1. '(iii) The only homomorphisms of –ö are the maps <!>‚Äě{z) = z", n e /.
(iv) The only homomorphisms of K" to –ö are maps of the form
(–≥,------‚Äě)1‚ÄĒ*‚Ė† .....where mu ... ,mne Z.
Prooi-'. Lei d denote the usual Huclidean metric on –ö which is a rotation invariant metric on K.
(i) Let II be a closed subgroup of K. If I! is infinite it has a limit point so Vf,> 0 3a, b e II with d{a, b) < t: and –į –§ h. Then d(b~la, 1) < e, and therefore the powers of h~ la are c-dcnsc in K. Therefore H is c-dense in –ö and H = K.
If H is finite and has p elements then ap = 1 Va e II. So each clement of H is a />-lh root of unity, and sinus there arc p elements in II, H must consist of all the p-th roots of unity.
(ii) Let 0:K -* –ö be an automorphism. We have <7(1) = 1. Since ‚ÄĒ 1 is the only element of –ö of order 2 we have 0(‚ÄĒ 1) = ‚ÄĒ 1. Since i, ‚ÄĒ i are the only elements of order 4 either 0(i) = / and 0(‚ÄĒ i) = ‚ÄĒior0(i) = ‚ÄĒ t and 0( ‚ÄĒ i) = i. Consider the first —Ā–į—Ć—Ā. Since 0 maps intervals to intervals, the interval [l.ij from 1 to i is either mapped to itself or to [i, 1] (all intervals go anticlockwise). But since [1, does not contain ‚ÄĒ 1 it cannot be mapped to [i, 1 j so = [1, /]. The only element_of –ĺ der 8 in [1,/] is e7"/4‚Äô and so this must be fixed by 0. Therefore ^[l,c'‚ÄôI,/4] = [l,c'"l/4j By induction one shows that 0(e2lu,2k)
‚ÄĒ e2n,l2'‚Äė for cach –ļ > 0. It follows tha.. 0 fixes all the 2*-th roots of unity V/c > 0 and hence is the identity. In the second case one shows that 0{e2mi2'‚Äė) = 2 ‚ÄĒ 2m,2– —É–ī. > Q ,UK| ]1ciice = Z_1, Z S K.
(iii) Let Q:K -¬Ľ –ö be an endomorphism. If 0 is non-trivial, its image, 6(K), is a closed connected subgroup of –ö and so 0(K) = –ö by (i). The kernel Ker
0 is a closed subgroup of –ö so either Ker 0 = –ö or Ker 0 = Hp, the group of all p-th roots of unity, for some p. Tne first case corresponds to trivial 0. If Ker 0 = //,, let av\–ö/–ü., -¬Ľ –ö be the isomorphism given by ap(zHp) = zp, and let/), .K/Hp -¬Ľ –ö be the isomorphism induced by –≤(0^–ė—Ä) = 0(z)).Then GjOtJ1 is an automorphism of –ö and by (ii) either 01 a‚Äú!(z) = i Vz e –ö or
jjll.X F.iKlonuirpliisms of Tori
15
0,0.', ‚Äė(-) = * 1 Vr ‚ā K. Hence either 0(z) = Oi(zHp) = 0^p l(zp) = :pV;eX or Q(c) = =7* Vz 6 K.
(iv) Let v,:X -¬Ľ K" be the homomorphism that imbeds –ö in the i-th component of K", i e. 7,(z) = (1,1,..., 1,1,1,... ,1), where z appears in the i-th component. If 0:Kn -* –ö is a homomorphism then 0 ¬į ys:K -¬Ľ –ö is an endomorphism and so 0 ¬į —É¬£¬Ľ) - zmi for some ;h, e Z by (iii). Hence Previous << 1 .. 2 3 4 5 6 < 7 > 8 9 10 11 12 13 .. 99 >> Next 