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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
Download (direct link): anintroduction1982.djvu
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{*.,} + {У„} = {(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„} = {y„} wheie y„ = xn+1 i.e., T(x0,xl,x2, -•-)) = (xux2,...). The one-Sided shift is a continuous transformation. The preimage under T of any point consists of к points.
(If we replaced the special space У by any compact metric space then we can clearly define the two-sided and one-sided shift maps with “state space”
У)
If one has о 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 У. 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 к x к matrix with аи {0,1} for all i, j. Let XA = = 1 VneZ}. In other words XA consists of all the bisequences (x„)" ^ 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\Хл 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 Xл = X. If A = I, the identity matrix, then XA consists of only к 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 = И-» {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 [2]]. 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 К and suppose it is positioned so that it is tangent to the real line, R, at 0 6 R. Consider the map x -»x/2 on R and let T:K~* К
be the map derived from this using stereographic projection. In other words T(N) = N, T(S) = S and if в e (—я/2, я/2) is the angle shown in the diagram then T maps the point of К cutting the line with angle в to the point of К cutting the line with angle tan-1(tan(0)/2}). So if хф {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-+ со if x ф S.
§5.2 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 “irreducibility” 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 {Тлх :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.
§5.2 Minimality
121
(i) T is minimal.
(ii) The only closed subsets E of X with ТЕ = 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 Ф 0 and ТЕ = E. II v e E then 07(x) с E so X = 0г(х) с E. Hence X = E.
(ii) => (iii) If U is non-empty and open then E = TnU is closed and ТЕ = E. Since E ф 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”U for some n e Z so T_nx e U and 0r(x) is dense in X. □
A subset E of X is called T-invariant if ТЕ = 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 -» 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 -» X has a minimal set.
Proof. Let S denote the collection of all closed non-empty T-invariant subsets of X. Clearly $ Ф 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’s intersection property.) Thus, by Zorn’s lemma, 6 has a minimum element. This element is a minimal set for T. □
Remark. I rgodicity h;is (lie properties:
(i) Ail ergodic transformation is “indecomposable" in the sense that it cannot be decomposed into two transformations (see §1.5).
(ii) Every measure-preserving transformation on a decent measure space can be decomposed into ergodic pieces in a nice way.
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