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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 84 85 86 87 88 89 < 90 > 91 92 93 94 95 96 .. 99 >> Next —É - {y‚Äû}- oo- F¬∞r / e C(X, R) and –∏ > 1 let
1
var‚Äû(/) = sup{|/(x) - f(y)11x,yeX; xt = y, when ‚Äî (n ‚Äî l) ^ / < n ‚Äî l}.
Since / is continuous we know var‚Äû(/) -+ 0. Let us suppose/ has the stronger requirement that var‚Äû(f) goes to 0 at an exponential rate i.e. 3C > 0, a e (0, l) with var‚Äû(/) < Co.‚Äù Vn > l. (If f(x) only depends on a finite number of coordinates of x then / satisfies this assumption.) It can be shown that / has a unique equilibrium state pif and that the measure-preserving transformation T on the probability space (X, –©–•), pif) is a Bernoulli automorphism. Also if f,g 6 C(X,R) both satisfy the exponential condition then pif = pig iff 3c g R and h g C(X, R) with / ‚Äî g = c + h¬∞T ‚Äî h. This tells us many differ-
\$9.5 Equilibrium Suie&
—ç n
ent measures are characterised a¬ª unique equilibrium states livs pivcs a den^e subset of C(X,R) each member of which has a unique equilibrium state. We shall prove a special case of thib which generalises Theorem –ö.–ß.
Theorem 9.16. Let T.X-+X be the two-sided shift homeomorphism of the
space X = [j\ –£, –£ = {0, 1,... ,k - 1}. Let a0,al....., e R and
define f e C(X,R) by /(v) = aXo where x - {*‚Äû}?*,. Then f has a unique equilibrium state which is the product measure defined by the measure on Y which gives the point i measure
–∫- 1
j=0
Proof. Let ¬£ = {/40,... ,.4k_ j} denote the natural generator i.e At =
|Xq = /}. Weknow –∫–π(–¢) = h^(T,0 <. //‚Äû(¬£) V> ‚Ç A/(.Y, T)(Theorem 4.17). We know from the end of ¬ß9.1 thet P(T,f) = log(X*=o e"Jl Also Mf[X,T) —Ñ 0 by Theorem 9 13(iv). Let —Ä–µ Mf{X,T). Put p,f = —Ä(–ê–¶,
0 ¬£ i < –∫ ‚Äî 1. Then
–∫ - 1 –∫-1 ^ –ò–î) + X ajPj = X pMj ~ l0sP'i
–æ j=o
^ log —É ¬£ ea^j bv Lemma 9.9.
By Lemm? 9.0 we must have
e*
Pi ~ –∫ - 1 j~0
Also hJJ~) - HJ.¬£) so since
hJtT) = h(T, {) * I ¬£ –ù–î) (Theorem 4.10)
we have Hu(\JlZl T~‚Äò{) = nHJQ. Theorem 4.4(ii) implies —Ü is a product measure, and therefore p. is the product measure which gives At measure
228
9 Topological Pressure and Its Relationship with Invariant Measures
Remarks
Let T.X-* X be the two-sided shift homeomorphism as in Theorem 9.16.
(1) There exist f e C(X,R) which have more than one equilibrium state (for a nice description of this sec Hofbauer ).
(21 U/ eC<x-R)M/(A', T) is a dense subset of M{X, T) for the norm topology on C(X,R)* (and hence in the weak*-topology) (Israel , p. 117; Ruelle
, p. 52).
(3) If —Ü1–≥..., —Ü‚Äû e E(X, T) there exists / e C(X,R) with {p,.....p‚Äû} c
Mf(X,T) (Israel , p. 117; Ruelle , p. 52). Therefore every ergodic measure for the shift is an equilibrium state. This statement is not true for an arbitrary homeomorphism T.
(4) See Ruelle  for the connection of topological pressure and equilibrium states with the corresponding notions in physics. Ya. G Sinai was the first to use equilibrium states to study diffeomorphisms (see ¬ß10.1).
CHAPTER 10
Applications and Other Topics
In this chapter v,e briefly describe some applications of the concepts introduced in the earlier chapters and mention some iopics we have not discussed.
¬ß10.1 The Qualitative Behaviour of Diffeomorphisms
In the subject of differentiable dynamics one tries to understand the behaviour of T" for large n when T:M -* M is a diffeomorphism of a compact differentiable manifold M (see Smale ). Therefore one would like to know about the orbits (T"(x)|ne Z} of a large sot of points xe M. There is a natural notion of set of measure zero in M\ ‚Äî –∏ Borel subset A of M has smooth measure zero if the intersection A n U with every coordinate chart V has zero Lebesgue measure (i.e. if —Ñ: U -* Rp is the coordinate map then —Ñ(–ê n V) has Lebesgue measure zero in Rp). So it would be natural to try to understand the orbits of a set of points whose compliment is a set of smooth measure zero. It turns out that this problem is closely connected to the study of equilibrium states of
For a certain class of diffeomorphisms the following result
holds (see Bowen ). Let T:M -¬ª M be an Axiom A diffeomorphism of a compact manifold M. There is a finite collection {/*,,... ,—Ü–≥) of members of M(M, T) for which the following statements hold.
(i) The set
has positive smooth measure for each j, 1 <; j <; r, and i Bj has smooth
measure zero.
229
230
10 Applications and Other Topics
(ii) There is a natural function —Ñ e C(M.R) such that {/i,,... ,—Ü,} arc . exactly the ergodic equilibrium states for —Ñ.
The condition (111) Yj=o flj means that for every f \$C{M.R)
(1 ") X"To f(T‚Äòx) -* j /f//<j so that the average value of cach "observable*'
/ on ihe orbit of x is calculable by the measure So this result connects the study of equilibrium states of T to the understanuing of the asymptotic behaviour of the orbits of most points in XI D. Ruelle has suggested a programme of extending this, and related results to non Axiom A ciflcomor-phisms (see Ruelle , , ) and even to the case of infinite dimensional manifolds in an attempt to give a description of hydrodynamic turbulence. At the basis of these extensions are two ergodic theorems which we describe in the next section. Previous << 1 .. 84 85 86 87 88 89 < 90 > 91 92 93 94 95 96 .. 99 >> Next 