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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 53 54 55 56 57 58 < 59 > 60 61 62 63 64 65 .. 99 >> Next Theorem 5.24. Let T:X -* X be an expansive homeomorphism of a compact metric space. Then there is an integer –ļ > 0, a closed subset ¬£2 of
Xk = II {0,1,... –Ē- 1}
‚ÄĒ –ě–°
such that SQ = Q, where S is the shift on Xk, and a continuous surjection —Ą-.Q-* X such that
—Ą–Ď(—É)= –Ę—Ą(—É) ye ¬£2.
i
Proof. The proof will resemble that of Example 2 above. Let 8 be an expansive constant for T. Wc'shall construct a cover —É = {C0,..., Ct_,} of X by closed sets with diam (C.) < 8 for each i, –°) n Cj = n 8Cj if i –§ j and IJfJo cCj having no interior.
We can do this as follows. Take an open cover {B0,..., ¬£^-1} by open balls of radius 8/3. Let C0 = B0, and for n > 0 let C‚Äě = B‚Äě\(B0 u ‚Ė† ‚Ė† ‚ÄĘ u –Ē Then if i < j we have
C; n Cj = dCj n Ct (since int(Cj) is Bj\(B0 u ‚Ė† ‚ÄĘ ‚Ė† –ł Bj_ t))
= dCj n SCi (since SCj n int(Ci) a Bt\(B0 u ‚ÄĘ ‚ÄĘ ‚ÄĘ u t) = 0).
Also lJt=o 3C, —Ā Ui=o SBi which has no interior.
¬ß5.6 Expansive Homeomorphisms
143
L et D = (J- = o SCi and D rj = (J‚Äú–∂ T'D. Then is a first category set so X\b, is dense in X. For each x e X\Dx we can assign, uniquely, a member of Xk by x -¬Ľ(¬ę‚Äě)!-,‚Äě tf T"x e C‚Äěn. Let A denote the collection of points of Xk arising in this way and let ij/'.XXD^ -* –õ denote the map just defined. We want to show that —Ą is injective and that the inverse of —Ą can be extended to a continuous map —Ą: –õ -* X. This will follow if we show for each e > 0 there is an integer N such that whenever x, —É e X\De'and (^lx))‚Äě = (^(y))‚Äě for all |n| < N then t/(x, y) < ¬£.
Let e > 0 be given. Choose N so that diam (\/*= -n T"y) < e, by Theorem 5.23. If (^(–Ľ))‚Äě = (^(y))‚Äě for |n| < N then x, —É are in the same element of V - n T"y and so d(x, y) < e.
Since —Ą–Ę = –Ď—Ą we have —Ą–Ď(—É) = –Ę—Ą(—É) Vy g –õ. The map —Ą is surjective since the dense set X\DK is in its image. ‚Ė°
The following gives many measure-theoretic generators for an expansive homeomorphism.
Theorem 5.25. Let T be an expansive homeoinorphism of a compact metric space (X,d) and let 8 be an expansive constant for T. If ¬£ = {A u , Ak] is a partition of X into Borel sets with diam (Aj) < S, 1 < j < k, then V¬ę= - oo –Ę~–Ņ—Ź/(¬£) = –©–•). Therefore, if —Ü is a probability measure on (–•,–©–•)) for which T is measure-preserving then h(T) = h(T,sf(¬£)) (by Theorem 4.17).
Proof. Consider any open ball B(x,e). By Theorem 5.23 for each n > 1 choose Nn such that diam(\/{^_Nii T~'¬£) < l/–ł. Let En denote the union of all the members of T~% that intersect B(x; e ‚ÄĒ 1 /n). Also
>/*
B(x\ –≥. - 1/–ł) <= E‚Äě —Ā B(x; c) so (J E‚Äě = B(x; c).
–ü- 1
Therefore B(x; e) e \/‚Äú= _ ‚Äě T~ns/(¬£), and since every open set is a countable union of open balls we see each open set belongs to \/–Ņ=-¬ģ ^"‚ÄĘ–Ļ–≥–ß¬£)-Hence –©–•) = \/¬ę _ ‚Äě –Ę~–Ņ–Ľ/(^). ‚Ė°
The above result will be important in Chapter 8.
Let us examine some examples.
(1) Isometries are never expansive except on finite spaces. Therefore rotations on compact metrisable groups are not expansive if the group is infinite.
(2) Let A be an automorphism of the –ł-torus, and [/1] the corresponding matrix. Then A is expansive iff [A] has no eigenvalues of modulus 1.
Sketch of Proof. One first shows that A is expansive iff the linea: map A of Rn (that covers A) is expansive. (The definition of expansiveness does not need a compact space.) Then show that A is expansive iff the complexification
144
5 Topological Dynamics
of A is expansive. Then one shows that the complexification of A is expansive iff the transformation given by the Jordan normal form is expansive. Lastly, one shows that the normal form is expansive iff there are no eigenvalues of modulus 1.
(Note: By Theorem 5.25 any partition of Kn into sufficiently small –ł-rectangles is a measure theoretic generator for an expansive automorphism of Kn.)
(3) The two-sided shift on –ļ symbols is expansive, (and by Remark 1 so are all two-sided topological Markov chains).
Proof (1). Let the state space be {0, 1,..., –ļ ‚ÄĒ 1}. Let At = {{x‚Äě}:x0 = i}, i =0,1,..., –ļ ‚ÄĒ 1. Then A0 u u ‚Ė† ‚ÄĘ ‚ÄĘ u Ak-1 = X and each At is open. The cover a= {–õ0,..., –õ*_,} is a generator for the shift since if x e –ü-¬Ľ where the Ain e a then
*
X ‚ÄĒ (. . . , /_2, l_i. *0¬Ľ *1¬Ľ ‚Ė† ‚ÄĘ ‚ÄĘ)‚Ė†
We then use Theorem 5.22 ‚Ė°
Proof (2). Let d be the metric given by:
<–ß–ú.–ę)- –Ā –ö,?/"1 ‚Ė†
–ė= - –ĺ—Ā
Suppose {x‚Äě} # {y‚Äě}. Then for some n0, —Ö–Ņ–ĺ —Ą yno and
4(—ā–Ľ–ĺ{—Ö–Ņ},—ā–Ņ¬į{—É‚Äě}) = ¬£ i-K+‚Äě0-^+J
n‚ÄĒ - 00 ^
^ |—Ö–Į—Ä –£—Ź()| ^ ^ ‚ÄĘ
Thus 1 is an expansive constant. ‚Ė°
The last two examples show an expansive homeomorphism can have a dense set of periodic points. There are expansive homeomorphisms with no periodic points: in fact, there are expansive minimal homeomorphisms which can be chosen of the form T\L where ¬£ is a minimal set for an expansive homeomorphism T.
Expansiveness is not related to topological transitivity or the size of the non-wandering set. There is some restriction though on periodic points as the next result shows. Previous << 1 .. 53 54 55 56 57 58 < 59 > 60 61 62 63 64 65 .. 99 >> Next 