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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
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(6) If Г is a subgroup of G then H = {g e G|y(g) = 1 Vy e Г} is a closed subgroup of G and (б/fl) = Г. Notice it makes sense for elements of Г 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 Ф G there exists а у e 6, у ф 1 such that y(h) = 1 Vlie H. (We shall write this у(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 и so that y(a) Ф 1 we have §y(x)dm(x) = 0. □
(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/ — ayy ]2 < г.. Only a countable number of ay are non-zero. We have £./ec |ау|“ = ||/ \\ < cc. This representation of / is called the Fourier series of /. When G = К = (te С11;| = 1} the Fourier series of / is the classical Fourier series /(z) = ^ a„z" since G consists of the maps y(z) = z",ne Z.
(10) If A :G -*■ G is an endomorphism we can define the dual endomorphism A\G -> G by Ay — у ° A, у 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.
О Preliminaries
§0.8 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 «-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”) и-* (x,,. .. ,x„) + Z".
Theorem 0.14
(i) Ererv closed subgroup of К is either К nr is a finite cyclic group consisting of all p-tli roots of unit\ for some integer p > 0.
(ii) The only automorphisms of К are the identity and the map zi—>r-1. '(iii) The only homomorphisms of К are the maps <!>„{z) = z", n e /.
(iv) The only homomorphisms of K" to К are maps of the form
(г,------„)1—*■ .....where mu ... ,mne Z.
Prooi-'. Lei d denote the usual Huclidean metric on К 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 а Ф 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 К 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 -* К be an automorphism. We have <7(1) = 1. Since — 1 is the only element of К of order 2 we have 0(— 1) = — 1. Since i, — i are the only elements of order 4 either 0(i) = / and 0(— i) = —ior0(i) = — t and 0( — i) = i. Consider the first саьс. 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 — 1 it cannot be mapped to [i, 1 j so = [1, /]. The only element_of о der 8 in [1,/] is e7"/4’ and so this must be fixed by 0. Therefore ^[l,c'’I,/4] = [l,c'"l/4j By induction one shows that 0(e2lu,2k)
— e2n,l2'‘ for cach к > 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'‘) = 2 — 2m,2 уд. > Q ,UK| ]1ciice = Z_1, Z S K.
(iii) Let Q:K -» К be an endomorphism. If 0 is non-trivial, its image, 6(K), is a closed connected subgroup of К and so 0(K) = К by (i). The kernel Ker
0 is a closed subgroup of К so either Ker 0 = К 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\К/П., -» К be the isomorphism given by ap(zHp) = zp, and let/), .K/Hp -» К be the isomorphism induced by в(0^Ир) = 0(z)).Then GjOtJ1 is an automorphism of К and by (ii) either 01 a“!(z) = i Vz e К or
jjll.X F.iKlonuirpliisms of Tori
0,0.', ‘(-) = * 1 Vr K. Hence either 0(z) = Oi(zHp) = 0^p l(zp) = :pV;eX or Q(c) = =7* Vz 6 K.
(iv) Let v,:X -» K" be the homomorphism that imbeds К 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 -* К is a homomorphism then 0 ° ys:K -» К is an endomorphism and so 0 ° у£») - zmi for some ;h, e Z by (iii). Hence
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