# An introduction to ergodic theory - Walters P.

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By its definition, a minimal transformation is “indecomposable”. We know that each homeomorphism T:X -* X has a minimal set. However, in general, one cannot partition X into T-invariant closed sets Ea such that X = TEX = £aVa, and T'|£a is minimal (although we can in some

important cases). If T has such a decomposition it is sometimes called semisimple. An example of a transformation not admitting such a decomposition is an ergodic automorphism of a compact metric group. This is because there are some points x with 0r(x) dense and some points where 0r(x) is not dense. We shall see this in §5.4.

As one might expect a minimal transformation can have no invariant non-constant continuous functions.

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5 Topological Dynamics

Theorem 5.3. If T:X-*X is minimal homeomorphism and f e C(X) then f о T = / implies f is a constant.

Proof. Since / ° T = / we have / о Tn = / Vn e Z, so if we pick some x e X we know / is constant on the dense set 07(x). Since / is continuous it must be constant. □

Remark. It is clear from the above proof that the conclusion of the theorem is true if there is some point with a dense orbit (rather than all points having a dense orbit), so the conclusion of Theorem 5.3 does not characterise minimality.

Since a minimal transformation cannot have non-trivial closed invariant sets it will not have any finite invariant sets unless X is finite. The points of finite invariant sets are called periodic points:

Definition 5.3. If T:X -» X is a homeomorphism then x is a periodic point of T if T'x = x for some n ^ 1. The smallest such n is the period of x. A periodic point of period one is called a fixed point.

As we mentioned above, if X is infinite no minimal homeomorphism of X can have any periodu points. However we shall see that there are homeo-morphisms of X with a dense set of periodic points and with {x e X10T{x) is dense} also being dense. An ergodic automorphism of a finite-dimensional torus will have this property.

We now check whether the examples mentioned in §5.1 are minimal or not.

(1) The identity map of X is minimal iff X consists of a single puint.

(2) 1 hco.cm 5.4. Let G be a compact metric group and T(x\ = ax. Then T is minimal iff {a"'n e Z} is dense in X.

Proof. Let e denote the identity element of G. Since 0T(e) = {an:n e Z}, the minimality of T implies {a”:n e Z} is dense. Now suppose the powers of a are dense. Let x e X. We must show that 0г(х) = X. Let у e X. There exists И; such that an‘ -*■ yx~i, so that

T"'(x) = an‘ ■ x -* y.

Therefore, 07 (x) is dense in X. □

(3) An automorphism A of a compact metric group G is minimal iff G = {e). This is because A(e) = e.

(4) For affine transformations of compact metric groups necessary and sufficient conditions for minimality are known. For example, if G is also abelian and connected then T = a ■ A is minimal iff

00

Пте = {е} and [a,BG] = G

n = 0

§5.3 The Non-wandering Set

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where В is the endomorphism of G defined by B(x) = x~1 • A{x) and [a, BG] denotes the smallest closed subgroup of G containing a and BG. This was proved by Hoare and Parry [1].

(5) The shift on к symbols is minimal iff к = 0. This is seen from (3) above.

(6) The north-south map of К is not minimal because the orbit of the point N is not dense.

§5.3 The Non-wandering Set

One basic d.fference between an ergodic rotation of К and the north-south map of К is that the points of К have a recurrence property for ergodic rotations (if x e К then points in the orbit of x come very close to x, since the orbit is dense) whereas the points of K, except for N and S, do not have a recurrence property for the north-south map (If x Ф N then Tn(x) -* S). This difference motivates the definitions of this section.

Definition 5.4. Let T: X -► X be a continuous transformation of a compact metric space and let x e A'yThe оз-limit set of x consists of all the limit points of {T"x|n > 0} i.e. co(x) = {y e X\3nt ? oo with Т"‘(х)-> y}.

Theorem 5.5. Let T:X -*■ X be a cohtinuous transformation of a compact metric space and x e X. Then

(i) cu(x) Ф 0.

(ii) c)( x) is a closed subset of X.

(iii) 7Vo(.v) = u>(x). ^

Proof, (i) is clear.

(ii) Let yk e a>(x) for к > 1 and yk-* ye X. We want to show у e co(x). For each j > 1 choose kj with d(ykj, y) < \r Now choose ni with d(T"Jx, ykj) < \J and so that ns< nJ+1 for all j. Then d{T"Jx, y) < l/j so у e co(x).

(iii) It ie clear that Tco(x) c cj(x). Let у e co(x) and suppose Tn,x -»y. Then {Tn, _1(x)} has a convergent subsequence so Tn‘j~1(x) -* z for some ze X. Then rn,j(x) -*• T(z) so that T(z) = y. Since z e a>(x) we have Tu>(x) = a>(x).

□

Remarks

(1) If T is not a homeomorphism then T~1ai(x) can be larger than a>(x). We can see this for the one-sided shift by choosing x to be the point {x„}“cc with x„ = 0 for all n. Then to(x) = x but T~ 1x has к points.

(2) If T is a homeomorphism then we can define the co-limit sets for T~1 and these are called the а-limit for sets for T. Hence a(x) = {z e X13nf oo with T-n‘(x) -»z}.

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