AgeneralizedRuellePerron-Frobeniustheoremandsomeapplications P W

Astérisque

P ETER W ALTERS

A generalized Ruelle Perron-Frobenius theorem and some applications

Astérisque, tome 40 (1976), p. 183-192

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Société Mathématique de France Astérisque 40 (1976) p.183-192

A G E N E R A L I Z E D RUELLE P E R R O N - F R O B E N I U S T H E O R E M A N D SOME A P P L I C A T I O N S

Peter W a l t e r s

W e show how some p r o b l e m s on u n i q u e n e s s of e q u i l i b r i u m states and e x i s t e n c e of invariant m e a s u r e s can be deduced from a t h e o r e m about P e r r o n - F r o b e n i u s o p e r a t o r s .

Let (X,d) be a compact m e t r i c space. Let T : X — x be a c o n - tinuous surjection. W e shall assume T satisfies the following c o n d i t i o n s a ) , b ) , and c ) .

a) T is p o s i t i v e l y e x p a n s i v e , ie. 3 6>o such that

d ( T n x , T n y ) _< 6 ^Vn>o implies x = y. An e q u i v a l e n t d e f i n i t i o n is to r e q u i r e the e x i s t e n c e of an open cover {A^,...^^.} of X for w h i c h

00

f\ T nA. is either empty or a one point set for all c h o i c e s of the n=0 1n

sequence {in} 1 £ in £ k • C l e a x l y for each xeX the set T "^x contains at m o s t k p o i n t s .

b) T is a local h o m e o m o r p h i s m (ie. V x ^ X ^ an open n e i g h b o u r h o o d U of x so that TU is open and T : U — * T ( U ) is a h o m e o m o r p h i s m . )

c) For sufficiently small 6 , d(x,t) = 6 = > d(Tx,Ty) >_ 6 .

Let eQ be chosen so that

i) e0 is an e x p a n s i v e constant for T ,

ii) VxeX the ball B (x) of radius c and centre x is so that eo °

TB£ (x) is open and T:Be (x)—^TBe (x) is a homeomorphism and o eo o

iii) Condition c) holds whenever 6 _< eQ .

Examples of transformations satisfying a ) , b ) , c ) . 1. Subshifts of finite-type.

r i oo r ,00. 1 Here one can take the metric d({x^J- , iy^j-J55 V X T

n o n o Js. i J.

if k is the least for which x^ ^ y^ . 2. Expanding maps. (Shub [ 9-]).

Here X is a compact manifold equipped with a Riemannian metric and T is differentiable and satisfies the property :

^ X>1 for which

||DTv|| > X ||vf vveUT.x xexA

Let C(X) be the Banach space of all real valued continuous functions on X , with the supremum norm. We can define for each (j)€C(X) a Perron-Frobenius operator X^ : C ( X ) — > C ( X ) by

<jL f(x) = E ^ f (y) . £ is linear and positive. The members of

^ yeT~" x ^

a subclass of these are particularly useful.

Let G(T) = (geC(X) | g>o and E ^ g C y ) = 1 V x e x } . y€T x

If (j) = log g then <£, f (x) = Z g(y) f (y) and we have .Log g _,

y€T x

^log g UT = id' where UTf = f °T •

Let M(X) denote the collection of all Borel probability measures on X and let M(T) consist of the T-invariant ones. In the weak*-topology the convex set M(X) is compact and M(T) is a

A RUELLE PERRON-FROBENIUS THEOREM

compact c o n v e x subset of M(X) . 3 w i l l denote the a- a l g e b r a of B o r e l subsets of X . A n interesting subset of M(T) is o b t a i n e d from the following

Lemma 1. (Ledrappier [ 5 ] )

Let g € G ( T ) and w e w r i t e f> instead of i . If m e M ( : log g

the following are e q u i v a l e n t : i) <£*m = m .

ii) m e M (T) and E (f . _ ) (x) = I g(z) f(z) a.e.m. V f e L ' d p zeT ^Tx

iii) m € M(T) and hm(T) + m ( l o g g) _> h^ (T) + y(log g) V y e M ( T ) (ie. m is an e q u i l i b r i u m state for log g )

A m e a s u r e satisfying these p r o p e r t i e s is called a g - m e a s u r e . If m is a g - m e a s u r e w e have

0 = hm(T) + m ( l o g g) .

(This says that the p r e s s u r e of log g is 0) .

S u p p o s e from n o w on that T also satisfies the following mix:

c o n d i t i o n :

d) V E > O 3 N > ° such that VxeX T~Nx is e-dense in X .

For < | ) € C ( X ) , e>o and n € Z+ let

var U , e ) = sup | | $ (x) - 4 (y) | d ( T 1 x , T 1 y ) < e) .

1 o<i<n-l J

W e then h a v e the following result.

T h e o r e m 2. (Keane [ 3 ] W a l t e r s [ll] )

CO

Suppose g £ G (T) and £ var (log g, e_) < oo for some e,<e n=l n l 1 - o T/zerc ^niog gf ^ p( f ) Vf 6 C( X ) . ( : ± denotes convergence in the supremum norm ) . y t/ze unique q-measure for T .

Theorem 3. (Bowen [ 2 ] , Ratner [ 7 ] , Walters [ll])

Let g Z?e as in theorem 2. The measure-preserving trans for­

mation (T,y) has a Bernoulli natural extension.

One can relate ^ to some ^\Qq g by a theorem first proved by Ruelle for the full 2-shift.

Theorem 4. (Ruelle [ 8 ] , Bowen [2] for the case of subshifts of finite type, Walters [ll])

00

Suppose <|> € C (X) and I var ((f),e) < » ^02? sowe e7^e

n=i n I 0 Then 3 X>o , v€ M( X ) , h e C ( X ) suc/z tfcat h>o , v (h) =1 ,

£ h = X h , X * v= X v and X, nf z> h. v (f) V f e C( X ) . f i r

00

Also h satisfies h (x) _< exp ( E var (<j),e )) h(y) k+1 n

whenever d(T1x,T1y) _< e1 o<i<k-l .

Remarks.

1 . X>o and v G M ( X ) are uniquely determined by the condition

«£*v = Xv »

2. One can define the pressure of T to be a function :C(X)—>R.

One has the variational principle

P U ) = sup I h ( T ) + y(<|>)

1 y£ M ( T ) L y J

(Walters [ 1 0 ] ) . We say m is an equilibrium state for (J) if

A RUELLE PERRON-FROBENIUS THEOREM

hm(T) m(4>) = PT(cf>)

C o r o l l a r y 5 .

Let cj) be as in theorem 4. T h e m e a s u r e y^ , d e f i n e d by

(f) = v(h.f) , is the unique e q u i l i b r i u m state for <p . y^ is the u n i q u e g-measure for g - e^h . T h e n a t u r a l e x t e n s i o n of (T,y .)' is

X.hoT * a B e r n o u l l i shift. u. is p o s i t i v e on non-empty open sets and

<P voT~n -> y^ in M(X) .

PT(c()) = log X = lim j log l n 1 . n->°° d)

oo r If ¥ € C ( X ) also has £ var °° then

n=l

y^=yy <=> <f)-y = foT - f + c for some f £ C (X) and c £ R .

A p p l i c a t i o n s .

1. A x i o m A d i f f e o m o r p h i s m s .

T h e s e r e s u l t s are described fully in B o w e n [ 2 ] . W e just state here two r e s u l t s w h i c h can be deduced using corollary 5 and the B o w e n - S i n a i theory of M a r k o v p a r t i t i o n s .

T h e o r e m 6.

Let fig be a basic set for an A x i o m A diffeomorphism T and let <f>€C(fl ) be Holder continuous (ie. | <|> (x)-<J> (y) | £ a d(x,y)

for some

s ^ T h e o r e m 7.

If <)>,¥ : fig — * R are both Holder continuous then Vicf) = y^ <=> (j)-^ = u o T - u + c

for some Holder

2. I n v a r i a n t m e a s u r e s for e x p a n d i n g m a p s .

H e r e X is a compact m a n i f o l d w i t h a R i e m a n n i a n m e t r i c and T: X — ^ X is d i f f e r e n t i a b l e and satisfies || DTv || _> X|v|| for all t a n ­ gent v e c t o r s v . H e r e X is a constant larger than 1 . T h e m e t r i c

d on X w i l l be the one obtained from J || . T s a t i s f i e s a) . b) . c ) . d ) . (Shub [9]) .

Let m be the n o r m a l i z e d R i e m a n n i a n m e a s u r e on X d e f i n e d by

|| || . W e are seeking an invariant m e a s u r e e q u i v a l e n t to m .

D T : T X —KTrp X is linear and w e can take its d e t e r m i n a n t using the X X IX

R i e m a n n i a n m e t r i c and so d e f i n e T'(x) = det(DxT) . D e f i n e (j> € C(X) by $ (x) = log 1^

|T' (x) |

* 1S C k-1 if T is Ck . W e w i l l assume T is C2 .

Lemma 8.

Let h e L ' f m ) and m(h) = 1 . Then

h . m € M ( T ) <=> o&h = h a.e. m .

(By h.m w e m e a n the m e a s u r e y defined by y( f ) = m ( h . f ) ) . Since ( j> £ C' ( X ) , and since for small e1

d ( x , y) < e < e 1 => d ( T x , T y ) _> Xd(x,y) , oo

w e get E var„ (•,£) < 00 . n=l

H e n c e w e can apply t h e o r e m 4 and C o r o l l a r y 5. N o t e that o6^m=m so that b y r e m a r k 1 v=m and X=l . So t h e o r e m 4 a s s e r t s the e x i s ­ tence of h € C( X ) w i t h m ( h ) = lf h>o °^n=n and «^^f => h.m(f)

¥ f e C ( X ) . By c o r o l l a r y 5 or lemma 8 w e know y = h.m 6 M ( T ) . So y is an i n v a r i a n t m e a s u r e e q u i v a l e n t to m . W e list o t h e r p r o p e r t i e s of y .

1. m T ~ n ->y in M( X ) (Corollary 5 ) .

A RUELLE PERRON-FROBENIUS THEOREM

2. (T,y) has a B e r n o u l l i n a t u r a l extension. (Corollary 5)

3. h (T) = J log | T' (x)| dy(x)

lim n-*oo

+ y(4))+ y(4))

T h i s is b e c a u s e PT((j>) = log X = o so 0 = h (T) + y(4)) so that h^ (T) = -y(cf>)

M d o g |T 1 I ) 4. m e M ( T ) <=> £ 1 = 1. V X 6 X .

y € T - i x |t' (y) |

5. S u p p o s e m C M ( T ) . T h e n m is the m e a s u r e w i t h m a x i m a l entropy

< = > |T' (x) | € Z+ V x 6 X .

M o s t of these results have been obtained by K r z y z e n s k i [ 4 ] .

3. M a p p i n g s of [0;l] .

Let T:[0,l] + [0,l] be a m a p satisfying

i) there is a p a r t i t i o n o=a <a.. <. . . <a =1 such that t| .

o l p ^ai'ai+i>

is C2 and can be extended to a C 2 function on tai ^ ai+il for each i .

ii) T m a p s each [ a . , a . , J 1 - 1 onto [0,l] .

1 p-1 iii) J X>1 such that | T ' ( x ) | >_ X V x € U (ai'*ai+1) •

J i=o E x a m p l e s are

aQ ax a2 a3 a0 al a2 a3

E a c h example defines a continuous m a p of S1 w h i c h is not smooth at a finite number of p o i n t s . In our first example w e could work as

for e x p a n d i n g m a p s but in the second example c o n d i t i o n s a ) , b) and c) are not satisfied. So w e proceed in the u s u a l w a y to u s e a shift system.

Let £ d e n o t e the p a r t i t i o n into the sets [ o , a 1 ) , [ a 1 , a 2 ) / . . . , [ a p _ 1 , l ] .

it ii ii

\ A2 A p

L e m m a 9.

£ is a g e n e r a t o r in the sense that each set of the form CO

fl T A . c o n t a i n s at most o n e p o i n t .

Let ft = { l , 2 , . . . , p } . D e f i n e j :[0,l] ft by j (x) = (x0,xlfx2, ) if T n x e A x j is 1-1 . j T = a j w h e r e a is the shift on ft .

00

Let Y = [0,1] \ U T~n{a ,a ,...,a } . T _ 1 Y = Y , a"1j(Y)=j(Y) . n=o p

Lemma 1 0 .

j is a h o m e o m o r p h i s m of Y w i t h j(Y^ .

j 1 e x t e n d s to a c o n t i n u o u s m a p ir:ft —*[0,l] it a = Ttt . Let m d e n o t e L e b e s g u e m e a s u r e on [ 0 , l ] . D e f i n e

V:Y — > R by Y(y) = log 1

| T F (y) |

L i f t Y to * = yon on j (Y) and this can b e e x t e n d e d to <j>ec(ft) .-1

w i t h E var (<b) < 00 . m is c o n c e n t r a t e d on Y so m°j d e f i n e s n=l

a m e a s u r e on j(Y) w h i c h d e f i n e s a m e a s u r e v on ft By d e f i n i t i o n of $ *t*v=v • By theorem 4 w e get

h > o h G C ( f t ) w i t h v(h)=l and o f £ f= > h v( f ) V f £ C ( f t ) . A l s o h . v 6 M ( a ) .

y= (h.v)OTr""1 G M ( T )

(h.v)oTT-1 = h o j . m on Y so y = £ . m € M ( T )

A RUELLE PERRON-FROBENIUS THEOREM

w h e r e £ is c o n t i n u o u s on Y and £>o . Hence T has an invariant m e a s u r e e q u i v a l e n t to m .

Of c o u r s e ( T, y ) is m e a s u r e - t h e o r e t i c a l l y i s o m o r p h i c to (a, h v ) and so has a B e r n o u l l i n a t u r a l e x t e n s i o n . The p r o p e r t i e s listed for e x p a n d i n g m a p s also hold in this c a s e . See Adler [l] for one of the sources of such r e s u l t s . T h e o r e m 4 can be extended so that one can handle the case w h e n ç is a c o u n t a b l e partition into i n t e r v a l s .

R E F E R E N C E S

[l] R.L. A d l e r , F - e x p a n s i o n s r e v i s i t e d . Recent A d v a n c e s in T o p o - logical D y n a m i c s . S p r i n g e r lecture n o t e s . V o l . 3 1 8 .

[2] R. B o w e n , E q u i l i b r i u m states and the E r g o d i c T h e o r y of A n o s o v D i f f e o m o r p h i s m s . S p r i n g e r l e c t u r e notes V o l . 4 7 0 .

[ 3 ] M. K e a n e , S t r o n g l y m i x i n g g - m e a s u r e s . Invent. M a t h . 16 (1972) 309 - 3 2 4 .

[4] K. K r z y z e w s k i , On e x p a n d i n g m a p p i n g s . B u l l , de l'académie P o l o n a i s e des sciences V o l . XIX N o . 1 1971.

[5] F. L e d r a p p i e r , P r i n c i p e v a r i a t i o n n e l et systèmes d y n a m i q u e s s y m b o l i q u e s . C o m m e n . M a t h s . P h y s . 33 (1973) pp 119 - 1 2 8 . [6] F. L e d r a p p i e r , M é c h a n i q u e statistique de l'équilibre pour un

r e v ê t e m e n t . P r e p r i n t .

[7] M. R a t n e r , A n o s o v flows w i t h G i b b s m e a s u r e s are also B e r n o u l l i a n . Israel J. M a t h . T o a p p e a r .

[8] D Ruelle, Statistical mechanics of a one-dimensional lattice gas. Comm. Math. Phys. 9 (1968) 267 - 278.

[ 9 ] M. Shub, E n d o m o r p h i s m s of compact d i f f e r e n t i a b l e m a n i f o l d s . A . J . M . XC1 Jan. 1969.

[lO] P. W a l t e r s , A v a r i a t i o n a l p r i n c i p l e for the p r e s s u r e of a con- tinuous t r a n s f o r m a t i o n . T o appear in A . J . M .

[ll] P. W a l t e r s , R u e l l e ' s o p e r a t o r theorem and g - m e a s u r e s . T o appear in T r a n s . A . M . S .

M a t h e m a t i c s I n s t i t u t e U n i v e r s i t y of W a r w i c k C o v e n t r y , W a r w i c k s h i r e CV4 7AL GB