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# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 43 44 45 46 47 48 < 49 > 50 51 52 53 54 55 .. 99 >> Next {*.,} + {–£‚Äû} = {(xn + >n)m¬∞d(/c)},
and T is an automorphism of X.
(6) There is a one-sided shift map corresponding to the two-sided shift in (5). If Y is as in example (5) then let X = Y be equipped with the product topology. The one-sided shift T:X X is defined by T{x‚Äû} = {y‚Äû} wheie y‚Äû = xn+1 i.e., T(x0,xl,x2, -‚Ä¢-)) = (xux2,...). The one-Sided shift is a continuous transformation. The preimage under T of any point consists of –∫ points.
(If we replaced the special space –£ by any compact metric space then we can clearly define the two-sided and one-sided shift maps with ‚Äústate space‚Äù
–£)
If one has –æ continuous map T:X -* X of a compact space and a clo^cd subset Y of X with TY a Y then T|y is a continuous map of the compact space –£. The map T|y is sometimes called a subsystem of T. There are many interesting subsystems of the shift maps of example (5). The following is one of them and is the topological analogue of a Markov chain.
(7) Let T:X X be the two-sided shift as in example 5. Let A = (fl^jLo be a –∫ x –∫ matrix with –∞–∏‚Ç {0,1} for all i, j. Let XA = = 1 VneZ}. In other words XA consists of all the bisequences (x‚Äû)" ^ whose neighbouring pairs are allowed by the matrix A. The complement of XA is clearly open so XA is a closed subset of X. Also TXA = XA so that T\–•–ª is a homeomorphism of XA and is called the two-sided topological Markov1 chain (or subshift of finite type) determined by the matrix A. For simnlicity we shall write T:XA -* XA rather than If (a,-j) = 1 all i,j then X–ª = X. If A = I, the identity matrix, then XA consists of only –∫ points. Sometimes XA is empty; for example when A = (? o) and X = {0,1}. Two matrices can define the same topological Markov chain; for example At = (o 1),
^2 = (o?) when X = –ò-¬ª {0,1}.
120
5 Topological Dynamics
Topological M irkov chains are very important as a source of examples and as models for important diffeomorphisms [see Bowen ]. One can also define one-sided topological Markov chains using example 6 rather than 5.
(8) This example is one of the simplest used in the qualitative study of diffeomorphisms of compact manifolds. It is called the north-south map. Consider the unit circle –ö and suppose it is positioned so that it is tangent to the real line, R, at 0 6 R. Consider the map x -¬ªx/2 on R and let T:K~* –ö
be the map derived from this using stereographic projection. In other words T(N) = N, T(S) = S and if –≤ e (‚Äî—è/2, —è/2) is the angle shown in the diagram then T maps the point of –ö cutting the line with angle –≤ to the point of –ö cutting the line with angle tan-1(tan(0)/2}). So if —Ö—Ñ {N,S} then T(x) is closer to S than x was and Tn{x) -* S as n -* oo. Also T~"x-* N as n-+ —Å–æ if x —Ñ S.
¬ß5.2 Minimality
In this section X will denote a compact metric space and T:X -* X a ho-meomorphism. We shall study homeomorphisms in this section and later we will consider continuous transformations. We would like to find a concept of ‚Äúirreducibility‚Äù to play the role ergodicity played for measure-preserving transformations.
Definition 5.1. A homeomorphism T:X-* X is minimal if Vx e A" the set {–¢–ª—Ö :n g Z} is dense in X. The set 0r(x) = {Tnx:n e Z} is called the T-orbit of x.
Theorem 5.1. The following are equivalent for a homeomorphism T:X-* X of a compact metric space.
¬ß5.2 Minimality
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(i) T is minimal.
(ii) The only closed subsets E of X with –¢–ï = E are 0 and X.
(ii.) For every non-empty open subset U of X we have TnU = X.
Proof
(i) => (ii). Suppose T is minimal and let E be closed, E –§ 0 and –¢–ï = E. II v e E then 07(x) —Å E so X = 0–≥(—Ö) —Å E. Hence X = E.
(ii) => (iii) If U is non-empty and open then E = TnU is closed and –¢–ï = E. Since E —Ñ X we have E = 0.
(iii) => (i). Let x e X and let U be any non-empty open subset of X. By
(iii) x e T‚ÄùU for some n e Z so T_nx e U and 0r(x) is dense in X. ‚ñ°
A subset E of X is called T-invariant if –¢–ï = E. If ¬£ is closed and T-invariant then T\E is a homeomorphism of the compact metric space E.
Definition 5.2. Let T:X -¬ª X be a homeomorphism. A closed subset E of X which is T-invariant is called a minimal set with respect to T if T\E is minimal.
Theorem 5.2. Any homeomorphism T:X -¬ª X has a minimal set.
Proof. Let S denote the collection of all closed non-empty T-invariant subsets of X. Clearly \$ –§ 0 since X belongs to S. The set 8 is a partially ordered set under inclusion. Every linearly ordered subset of & has a least element (the intersection of the elements of the chain. The least element is non-empty by Cantor‚Äôs intersection property.) Thus, by Zorn‚Äôs lemma, 6 has a minimum element. This element is a minimal set for T. ‚ñ°
Remark. I rgodicity h;is (lie properties:
(i) Ail ergodic transformation is ‚Äúindecomposable" in the sense that it cannot be decomposed into two transformations (see ¬ß1.5).
(ii) Every measure-preserving transformation on a decent measure space can be decomposed into ergodic pieces in a nice way. Previous << 1 .. 43 44 45 46 47 48 < 49 > 50 51 52 53 54 55 .. 99 >> Next 