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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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Conversely if T is weak-mixing and A is not ergodic then by (а) у » A = у for some уф 1. But then
|(^тУ,У)| =|(у(Ф(Ла) • • • у(Л""1а)}’,у)! = ||у|Ц = 1
for all n contradicting the weak-mixing of T. So if T is weak-mixing then A is ergodic. j □
(6) Theorem 1.30. The two-sided (p0,... ,pk^I)-shi/t is strong-mixing.
Proof. Use Theorem 1,17 after verifying the correct behaviour for measurable rectangles. □
(7) Similarly, the one-sided (p0,... ,pk_ J-shift is strong-mixing.
(8) We have the following theorem for the (p,P) Markov shift. Recall that we can always assume cach pt > 0.
Theorem 1.31. If T is the (p,P) Markov shift (either one-sided or two-sided) the following are equivalent:
(i) T is weak-mixing.
(ii) T is strong-mixing.
(iii) The matrix P is irreducible and aperiodic (i.e. 3N > 0 such that the matrix PN has no zero entries).
(iv) For all states i, j p\f -* pj.
Proof. That (iii) and (iv) are equivalent is a standard use of the renewal theorem in probability theory.
(i) => (iii). Since (1 /N) Ной о Tq"[/]0) - «i(0[i]0)m(0[/]0)| - 0 we have (1/N) Y£=o И;’ ~ Pj\ 0- By Theorem 1.20 we get a set J of density zero in Z+ such that
lim p{jj = pj for all i, j.
Therefore P is irreducible and aperiodic.
1 Measure-Preserving Transformations
(iv)=>(ii). Dv Theorem 1.17 it suffices to show that for two blocks A = «['o ’' ' /,]■*,. В = t[y0 '' 'лЗл-м we have т{Т~пА n B) — m(A)m(B). This is a straightforward calculation (see proof of Theorem 1.19 for a similar one).

Remark. We know that T is ergodic iff V/eL^mXl/n) о U'rf dm a.e. There is the following connection between strong-mixing and convergence of ergodic averages. The measure-preserving transformation T is strong-mixing iff for every increasing sequence /с, < k2 < ■ • ■ of natural numbers and every feL2(m) we have ||(l/n) U^'f — J/A»||2 -* 0 (Blum and Hanson [2]).
Isomorphism, Conjugacy, and Spectral Isomorphism
So far we have been studying measure-preserving transformations on probability spaces. We now want to consider the notion of isomorphism for measure-preserving transformations; in other words, when should we consider two measure-preserving transformations as being “the same” or being equivalent? We must bear in mind that in measure theory a set of measuie zero can be ignored. We first consider ways of ignoring sets of measure zero before considering isomorphism of measure-preserving transformations.
§2.1 Point Maps and Set Maps
One of the most important notions in measure theory is that of neglecting sets of measure zero. With this in mind let us consider what we should mean by saying two probability spaces (X,(X2,&2,m2) are isomorphic. One way to view this is to require that the spaces be connected by an invertible measure-preserving transformation after removing sets of measure zero from each space.
Definition 2.1. The probability spaces (X {X2,&2,m2) are said to
be isomorpftic if there exist Mx e M2 ь&$2 with m^Mj) = 1 =m2(M2) and an invertible measure-preserving transformation ф:МM2. (The space M; is assumed to be equipped with the c-algebra M; n Й, = {М,- n B\B в 3d,) and the restriction of the measure w, to this cr-algebra.)
There is the following theorem on isomorphism of measure spaces.
Theorem 2.1. Let X be a complete separable metric space, let 82(X) be its a-algebra of Borel subsets and let m be a probability measure on S3(X) with
2 Isomorphism, Conjugacy, and Spcctral Isomorphism
"'({-v}) = 0 for each set (л) consisting of a single point x e X. Let ([0,1], £$([0,1]),/) denote the closed unit interval with its o-algebra of Borel sets and Lehesgue measure I. Then (X, 3{X), m) and ([0, 1], 38([Q, 1]), /) are isomorphic. If (X,&3„,(X)jn) denotes the completion of (X,SS{X),m) then (X JSSm{X), m) is isomorphic to ([0, 1], J5?,/) where ££ is the o-algebra of Lebesgue measurable sets (which is the completion J?(([0, 1])).
A proof is given in Theorem 9, page 327 of Royden’s book (Royden [1]).
If the condition that in have no points of positive measure is1 omitted then there are at most a countable collection of points {*,,}" of X with positive measure and then (X,zft(X),m) is isomorphic to a measure space consisting of points {j1,,}!0 with measures {m(ic„)} together with ([0,.s], а?([0,^]),/) where s=l — ]T^L j m(xn). The corresponding statement for completed spaces is true. Therefore all the probability spaces on which our examples act come under these results.
There is another way to handle the omission of sets of measure zero, and this other way is, perhaps, mathematically more natural but less practical in applications. This is the idea of using measure algebras.
Let (X,a?,ni) be a probability space. Define an equivalence relation on £8 by saying A and В are equivalent {A ~ B) iff m(A Д B) — 0. Let 36 denote the collection of equivalence classes. Then 28 is a Boolean ст-algebra under the operations of complementation, union and intersection inherited from
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