CONVERGENCE OF THE RUELLE OPERATOR FOR A FUNCTION SATISFYING BOWEN S CONDITION

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 1, Pages S (00) Article electroically published o September 13, 2000 CONVERGENCE OF THE RUELLE OPERATOR FOR A FUNCTION SATISFYING BOWEN S CONDITION PETER WALTERS Abstract. We cosider a positively expasive local homeomorphism T : X X satisfyig a weak specificatio property ad study the Ruelle operator L of a real-valued cotiuous fuctio satisfyig a property we call Bowe s coditio. We study covergece properties of the iterates L ad relate them to the theory of equilibrium states. 1. Itroductio We cosider a cotiuous map T : X X of a compact metric space ad a cotiuous fuctio : X R with some assumptios that esure the Ruelle operator L maps the space C(X; R) of real-valued cotiuous fuctios to itself ad behaves well. We obtai a covergece theorem for L as (Theorem 2.16) ad deduce results about equilibrium states. Results of this type are mostly stated whe T is a subshift of fiite type, but we use a more geeral cotext to iclude expadig maps of compact maifolds ad other examples. Let X be a compact metric space with metric d, ad let T : X X be a cotiuous surjectio. We shall assume T is positively expasive, i.e. δ 0 > 0, so that if x z 0withd(T x, T z) > δ 0. Such a umber δ 0 is called a expasive costat for T ad clearly every smaller positive umber is also a expasive costat. If we chage to a equivalet metric, the T is positively expasive i the ew metric, but the expasive costat ca chage. Reddy has show oe ca fid a equivalet metric D ad costats τ>0, > 1 such that D(x, z) <τ implies D(Tx, Tz) D(x, z). Hece T expads distaces locally i the metric D ([Re]). We also assume T is a local homeomorphism. This coditio ca be stated i several equivalet ways, which we discuss later i this sectio. The third assumptio o T is a weak specificatio coditio, which ca also be described i several ways (see Theorem 1.2). This coditio does ot ivolve periodic poits. Topologically mixig subshifts of fiite type are examples of positively expasive local homeomorphisms with the weak specificatio property, ad they are the oly subshifts with these properties. Aother importat class of examples is give by expadig differetiable maps of smooth compact coected maifolds. We have assumed T is a local homeomorphism because we wat the trasfer operators, defied by Ruelle, to map the Baach space C(X; R) of real-valued cotiuous fuctios o X, equipped with the supremum orm, to itself. For Received by the editors August 9, Mathematics Subject Classificatio. Primary 37D35; Secodary 28D20, 37A30, 37B10. Key words ad phrases. Trasfer operator, equilibrium state, etropy. 327 c 2000 America Mathematical Society

2 328 PETER WALTERS each C(X; R) the trasfer operator L : C(X; R) C(X; R) is defied by (L f)(x) = y T 1 x e(y) f(y). This is a fiite sum; sice T is positively expasive, each set T 1 x is δ 0 -separated if δ 0 is a expasive costat. Each operator L is liear, cotiuous ad positive. We take the opportuity to itroduce some otatio. We use 1 to deote the costat fuctio with value 1. The ope ball with ceter x ad radius δ will be deoted by B(x; δ). The σ-algebra of Borel subsets of X will be deoted by B(X) orbyb if o cofusio ca arise. The covex set, M(X), of all probability measures o (X, B) ca be cosidered as a subset of the dual space C(X; R) ad M(X) is compact i the weak -topology. The space of T -ivariat members of M(X) is also compact i the weak -topology ad is deoted by M(X, T). If C(X; R), the P (T,) deotes the pressure of T at ([W2]). If µ M(X), the L p µ (X) deotes the space of measurable f : X R with f p itegrable with respect to µ, p 1. The coditioal expectatio of f : X R with respect to a σ-algebra A usig µ is deoted by E µ (f/a). We sometimes write µ(f) istead of fdµ. If C(X; R), the a equilibrium state for is some µ M(X, T) with h µ (T )+µ() =P (T,)whereh µ (T ) is the etropy of the measure-preservig trasformatio T :(X, B,µ) (X, B,µ). A equivalet coditio is h σ (T )+σ() h µ (T )+µ() σ M(X, T) ([W2]). Thesymbol deotes uiform covergece. If 1ad: X R, weuse(t )(x)for 1 i=0 (T i x). Note that (L f)(x)= y T x e(t)(y) f(y). For C(X; R), 1, δ>0 defie v (, δ) =sup{ (x) (z) d(t i x, T i z) δ, 0 i 1}. If δ is a expasive costat, the v (, δ) 0as. The followig result, i which L deotes the dual of L, is well-kow, ad the first versio of it was proved by Ruelle ad is called the Ruelle operator theorem. Theorem 1.1. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let C(X; R) satisfy sup v +j (T, δ) 0 as j for some δ>0 1 (ad hece for all smaller δ). There exists h C(X; R) with h>0, R with >0, ad v M(X) so that L h = h ad L ν = ν. If we ormalize h so that ν(h) =1,the f C(X; R) (L f)(x) h(x)ν(f). The spectral radius of L : C(X; R) C(X; R) is ad log = P (T,). Proof. The coditio o still holds if we chage to a equivalet metric. The result is a special case of Theorem 8 of [W1] because uder a Reddy metric D the map T satisfies the coditios of [W1]. Sice v +j (T, δ) j+ i=j v i(, δ), the coditio o is implied by =1 v (, δ) <. Oe cosequece of Theorem 1.1 is that each satisfyig the coditio of the theorem has a uique equilibrium state µ,givebyµ = h.ν (i.e. µ (f) =ν(h.f) f C(X; R)) ad T is exact with respect to µ. Also, uder the coditios of the theorem oe ca show the atural extesio of the measure-preservig trasformatio T :(X, B,µ ) (X, B,µ ) is isomorphic to a Beroulli shift ([W1]).

3 CONVERGENCE OF THE RUELLE OPERATOR 329 We wat to cosider a weaker coditio o tha the oe i Theorem 1.1. It is worth poitig out that oe eeds o assumptio o C(X; R) toget>0 ad ν M(X) withl ν = ν, as is easily see by usig the Schauder-Tychaoff fixed poit theorem o the map µ L µ (L µ)(1) of M(X). We shall cosider the followig coditio that resembles oe used by Bowe i the case of expasive homeomorphisms with a strog specificatio property ([B1]). Defiitio. We say C(X; R) satisfies Bowe s coditio (with respect to T : X X) if δ >0adC>0with the property that wheever d(t i x, T i z) δ for 0 i 1, the 1 [(T i x) (T i z)] C. i=0 Aother way to phrase this defiitio is sup 1 v (T, δ) <. If Bowe s coditio holds for some δ, the it holds for all smaller δ. Noticethatif C(X; R) satisfies the assumptio of Theorem 1.1, the it satisfies Bowe s coditio. There are examples of fuctios that satisfy Bowe s coditio but ot the assumptio of Theorem 1.1. Later results will show the Bowe coditio is a atural assumptio (Theorem 4.8). Bowe showed that whe T is a expasive homeomorphism with the strog specificatio property, the every cotiuous fuctio satisfyig his coditio has a uique equilibrium state ad this state is weak-mixig. He uses a method based o periodic poits. We cosider how much of Ruelle s operator theorem holds for C(X; R) satisfyig Bowe s coditio. We prove there is a uique ν M(X) withl ν = ν for > 0 ad this measure is tail-trivial. Recall that ν is tail-trivial meas that the oly values ν takesosetsithetailσ-algebra B = =0 T B(X) are 0 ad 1. Moreover, we show f C(X; R) (L f)(x) (L 1)(x) coverges uiformly to ν(f) (Theorem 2.16). We show has a uique equilibrium state µ ad ivestigate some properties of µ. This is doe by cosiderig measurable g-fuctios after we costruct a measurable h: X [d 1,d 2 ] (0, ) withl h = h everywhere. We use a method suggested by the work of Fa who studied the case of cotiuous g-measures for subshifts of fiite type ([F]). We also cosider covergece of L f as ad obtai a result givig L p covergece. Such a result has bee proved by Ruelle i a more geeral cotext i which the trasformatio T eed ot be a local homeomorphism ad hece the operator L does ot act o C(X; R) ([Ru]). Our measurable g-fuctio method allows us to deduce the L p covergece from a Martigale theorem. I 2 we prove the covergece theorem (Theorem 2.16) ad characterize the Bowe coditio i terms of measures. I 3 we deduce results about g-measures from the results of 2. We study equilibrium states ad the covergece of L f i 4 usig measurable g-fuctios. I 5 we cosider cotiuity properties of a measurable desity fuctio obtaied i 4. We ow discuss the assumptio that T be a local homeomorphism. The coditio ca be stated i several equivalet ways. Oe statemet, that is easier to check i examples, is that for every x X thereisaopeeighbourhoodu x of x with TU x ope such that T maps U x homeomorphically oto TU x. By a result of Eileberg ([A-H], p. 31) this is equivalet to the followig statemet i which

4 330 PETER WALTERS diam(b) deotes the diameter of the set B X: there exist δ 1 > 0, θ>0ada fuctio η :(0,δ 1 ] (0, ) with lim t 0 η(t) = 0 such that each ope subset V of X with diam(v ) <δ 1 has a decompositio of T 1 V with the followig properties: (i) T 1 V = U 1 U k for some k, whereeachu i is ope; (ii) T maps each U i homeomorphically oto V ; (iii) if i j, thed(x i,x j ) θ x i U i,x j U j ; (iv) diam(v ) < δ implies diam(u i ) < η(δ) for each i, 1 i k, ad each δ (0,δ 1 ]. If we chage to a equivalet metric, the this property still holds with differet δ 1,θ ad η. We ow show that if T : X X is a positively expasive local homeomorphism, there is some δ 2 > 0 such that wheever V is a ope subset of X with diam(v ) <δ 2 ad x V,the 1, T V is a disjoit uio y T x U y of ope sets with T mappig each U y oto V.ToseethischooseaReddymetricD with costats τ>0, >1, ad let δ 1,θ,ηcorrespod to D i the local homeomorphism coditio. Let δ satisfy δ<δ 1,δ<τad η(δ) <τ ad suppose V is ope ad diam D (V ) <δ (where the subscript shows the diameter is take for the metric D). The T 1 V = U 1 U k whereeachopesetu i has diam D (U i ) δ/ < δ. Therefore each T 1 U i ca be decomposed ito disjoit ope sets of D-diameter at most δ/ 2.By iductio we have 1 x V T V is a disjoit uio y T x U y of ope sets of D-diameter at most δ/ ad T maps each U y homeomorphically oto V icreasig distaces. If we ow revert to the origial metric d there is some δ 2 > 0 such that if V is ope ad diam(v ) <δ 2 ad x v, the 1 T V is a disjoit uio y T x U y of ope sets with T mappig each U y homeomorphically oto V. We ow cosider the coditio of weak specificatio o T. Several equivalet forms are give i the followig theorem i which B (x; ε) deotes the closed Bowe ball {y X d(t i x, T i y) ε, 0 i 1}. Theorem 1.2. For a positively expasive local homeomorphism T : X X the followig statemets are pairwise equivalet: (i) ε >0 N >0 such that x XT N x is ε-dese i X. (ii) ε > 0 M > 0 such that x, x X 1 w T (+M) x with d(t i w, T i x) ε, 0 i 1. (iii) ε >0 M >0 such that x X 1 T +M B (x; ε) =X. (iv) ε >0 M 0 such that x, x X 1, 2 1 w X with d(t i w, T i x) ε, 0 i 1 1 ad d(t 1+M+j 1 w, T j x 2 ) ε, 0 j 2 1. Proof. Note that each of the statemets is idepedet of the metric. We use B d (x; ε) for the ope ball with ceter x ad radius ε i the metric d. Assume (i) holds ad we prove (ii). Let D be a Reddy metric with D(Tx,Tx ) D(x, x ) wheever D(x, x ) < τ. Let δ 1,θ,η be associated to D by the local homeomorphism property. Let ε>0satisfyε< δ1 2, ε<τ/2adη(ε/2) <τ,ad let N correspod to ε i statemet (i) for the metric D. By statemet (i) choose y T N x with D(y,T x) <ε. By the discussio before the statemet of the theorem T B D (T x; ε) ca be writte as a disjoit uio w T T x U w of ope sets with D-diameteratmost2ε/ ad T maps each U w homeomorphically oto

5 CONVERGENCE OF THE RUELLE OPERATOR 331 B D (T x; ε). Let v = T y U x.thev T (+M) x ad for 0 i 1 D(T i xt i v ) D(T x, T v ) i <ε/ i. Hece statemet (ii) holds for the metric D, ad therefore for the origial metric d. Statemet (iii) is clearly the same as statemet (ii), ad it clearly implies statemet (iv). Now assume (iv) holds ad we prove (i). Let ε be a expasive costat ad x, z X. We wat to fid y T N x with d(z,y) <ε. By statemet (iv) M so that 1 w x with d(w,z) ε/2 add(t M+j w,t j x) ε/2 for 0 j 1. Choose a coverget subsequece w i w to get d(w, z) ε/2 ad d(t M+j w, T j x) ε/2 for all j 0. Hece T M w = x. Because of statemet (iv) we shall say that T satisfies the weak specificatio coditio if it satisfies oe, ad hece all, of the statemets i Theorem 1.2. If T satisfies the weak specificatio coditio, the T is topologically mixig. Topologically mixig subshifts of fiite type are examples of positively expasive local homeomorphisms with the weak specificatio property, ad they are the oly subshifts with these properties. Aother importat class of examples is give by expadig differetiable maps of smooth compact coected maifolds. For details about (, ε) spaig sets ad (, ε) separated sets see [W2]. Theorem 1.3. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. The C(X; R) 1 log(l 1)(x) P (T,). Proof. By weak specificatio if ε>0 M so that x X 1 T (+M) x is (, ε) spaig, by statemet (ii) of Theorem 1.2. If { } Q (, ε) =if e (T)(x) F is (, ε) spaig, x F the ε is a expasive costat P (T,) = lim if 1 log Q (, ε). So P (T,) lim if 1 (L+M 1)(x ) = lim if 1 log(l 1)(x ) x X. Sice T is positively expasive, T x is (, ε) separated if ε is a expasive costat. If P (, ε) =sup{ x E e(t)(x) E is (, ε) separated}, thep (T,) 1 lim sup log P (, ε), ad this is actually a equality if ε is a expasive costat. Therefore P (T,) lim sup (L 1)(x ), x X. Hece 1 1 log(l 1)(x) P (T,). 2. Covergece theorem We adapt the method of Fa ([F]) to our more geeral situatio. As before, let X be a compact metric space with metric d ad let T : X X be a positively expasive, local homeomorphism with the weak specificatio property. Let C(X; R) ad let its trasfer operator be L : C(X; R) C(X; R). Let

6 332 PETER WALTERS U T : C(X; R) C(X; R) be give by (U T f)(x) = f(tx). For 1 defie P () : C(X; R) C(X; R) by (P () f)(x) = (U T L f)(x) (UT L 1)(x) = (L f)(t x) (L 1)(T x) = z T T x e(t)(z) f(z). z T T x e(t)(z) These operators have the followig properties. Theorem 2.1. For C(X; R) ad 1 let P () be defied as above. (i) P () is liear, cotiuous ad positive. (ii) P () 1=1. (iii) If l C(X; R) is so that x Xliscostat o T x,the P () () (l f) =l P f f C(X; R). (iv) L P () = L. (v) If m, thep () P (m) = P (m) P () = P (m). (vi) f C(X; R) P () f is costat o each set T x. Proof. (i), (ii), (iii) ad (vi) are clear from the defiitio. To prove (iv) we have (L P () f)(x) = e (T)(y) (P () f)(y) y T x = y T x To prove (v) let m ad the P () P (m) e (T)(y) z T x e(t)(z) f(z) z T x e(t)(z) =(L f)(x). f = P (m) f P () 1 by (iii) ad (vi) = P (m) f by (ii); P (m) P () f = U T mlm P () f UT mlm 1 = U T mlm L P () UT mlm 1 f = U m T Lm f U m T Lm 1 by (iv) = P (m) f. The dual operator P () maps M(X) itom(x) ad is cotiuous for the weak - topology. The set K () = {ν M(X) P () ν = ν} is a compact covex set which is oempty by the Schauder-Tychaoff fixed poit theorem. Note that K () = P () M(X), by Theorem 2.1 (v). By Theorem 2.1 (v) we have K (1) K (2) ad K = =1 K() is a oempty, compact covex set. Let L deote the dual of L : C(X; R) C(X; R). Theorem 2.2. Let J : M(X) M(X) be defied by J µ = L µ (L µ)(1). The P () J = J ad JM(X) K (). Proof. By iductio Jµ = (L ) µ () ((L ) µ)(1).thep Jµ = Jµ by Theorem 2 (iv). Corollary 2.3. There exist ν M(X) ad >0 with L ν = ν. Every such ν is i K.

7 CONVERGENCE OF THE RUELLE OPERATOR 333 Proof. The measures ν M(X) withl ν = ν for some >0areexactlythe fixed poits of J.SiceJ has a fixed poit by the Schauder-Tychaoff theorem, the result follows from Theorem 2.2. We shall use the followig theorem to get iformatio about P () ad about. If we let L(X; R) deote the vector space of all Borel measurable fuctios K () f : X R, the we ca cosider L as a map L : L(X; R) L(X; R)adP () as a : L(X; R) L(X; R). We shall use the followig result about a operator map P () ad later whe P is equal to L. For realvalued fuctios the expressio f f meas that for each x, f 1 (x) f 2 (x) ad f (x) f(x). P i the cases whe P is equal to P () Theorem 2.4. Let P : L(X; R) L(X; R) be a liear trasformatio which restricts to a cotiuous liear operator P : C(X; R) C(X; R). Assume P is positive i the sese that if f L(X; R) ad f 0, thepf 0. Let P act o fiite measures by fd(p µ)= Pfdµ, f C(X; R). Assume that P also has the property that wheever {f } is a sequece i L(X; R) with f f poitwise, the Pf Pf poitwise. Let µ M(X). The for all f L(X; R) with f 0 we have Pf dµ = fd(p µ). I particular, B B(X)(P µ)(b) = Pχ B dµ. Also, if f L(X; R), thef L 1 P µ (X) iff Pf L1 µ(x) ad for such f we have Pfdµ = fd(p µ). Proof. Let C = {B B(X) Pχ B dµ =(P µ)(b)}. We have X, φ Cad C is closed uder complemets ad fiite disjoit uios. We show every ope set is i C. LetU be a ope subset of X ad write it as a coutable uio j=1 B(x j; r j )of ope balls. Let C = j=1 B(x j; r j 1 )whereb(x; r) is take as empty if r 0. Each C is closed, C 1 C 2 C 3 ad =1 C = U. By Urysoh s lemma choose a cotiuous f i : X [0, 1] with χ Ci f i χ U. The f i χ U poitwise so Pf i Pχ U. By the domiated covergece theorem Pf i dµ Pχ U dµ, ad, also by the domiated covergece theorem for P µ, f i d(p µ) χ U d(p µ). Hece U C. We ext show C = B(X). Defie a measure µ 1 o X by µ 1 (B) = Pχ B dµ, B B(X). The above shows µ 1 (U) =(P µ)(u) for all ope sets U, ad sice every fiite measure o X is regular, we have µ 1 = P µ. Hece C = B(X). If f is a oegative simple fuctio, the Pf dµ = fdp µ. If f is a oegative measurable fuctio, we ca choose a sequece {f j } of oegative simple fuctios with f j f. Sice P is positive we have Pf j Pf ad the desired result follows for f, adf L 1 P µ (X) iffpf L1 µ(x). By cosiderig positive ad egative parts of f L(X; R) wehavef L 1 P µ (X) iffpf L1 µ(x) ad for such f we have Pfdµ= fd(p µ). The followig result is ow clear. Corollary 2.5. If ν K () For f L 1 ν (X) we have fdν = P () ad f L(X; R), thef L 1 ν fdν. Corollary 2.6. If ν K (), f L 1 ν (X) ad B B(X), the fdν = P () fdν. T B T B () (X) iff P f L 1 ν (X).

8 334 PETER WALTERS Proof. By Corollary 2.5 fχ T Bdν = P () (f χ T B) dν = χ T BP () fdν by Theorem 2.1 (iii). The followig characterizes the members of K (). Corollary 2.7. For ν M(X) the followig statemets are equivalet: (i) ν K (). (ii) E ν (f /T B)=P () f a.e. (v) f L 1 ν(x). (iii) E ν (f /T B)=P () f a.e. (v) f C(X; R). Proof. We have (i) (ii) by Corollary 2.6. Clearly (ii) (iii). If (iii) holds ad f C(X; R), the P () fdν = E ν (f /T B)dν = fdν, so ν K (). So the elemets of K () are those probability measures with coditioal expectatio E(f/ T B)givebyP () f. Corollary 2.8. Let ν M(X) satisfy L ν = ν for 0. If f L(X; R) ad f 0 we have L fdµ = fdν. Also, for f L(X; R) we have f L 1 ν (X) iff L f L 1 ν (X) ad for these f, L fdν = fdν. Proof. Put P equal to L i Theorem 2.4. The followig result characterizes the situatio whe K is as small as possible. Theorem 2.9. Let T : X X be a positively expasive local homeomorphism with theweakspecificatiopropertyadlet C(X; R). The followig statemets are pairwise equivalet: (i) K has oly oe member. (ii) f C(X; R) c(f) R with P () (iii) f C(X; R) c(f) R with P () f c(f). f c(f) poitwise. (iv) f C(x; R) c(f) R with L f L 1 c(f). Whe these statemets hold the uique elemet ν of K satisfies L ν = ν for some >0 ad c(f) = fdν. Proof. (i) (ii). Let K = {ν}. If (ii) holds, the c(f) = fdν by the bouded covergece theorem. If (ii) fails, the f 0 C(X; R) ad ε 0 > 0 ad sequeces j ad x j X with (P (j) f 0 )(x j ) ν(f 0 ) ε 0 j 1. We ca write this (j ) f 0 d(p δ xj ) f 0 dν ε 0 1. Choose a coverget subsequece P (j) δ xj τ M(X). Sice K () = P () M(X), we have τ K. But f 0 dτ f 0 dν ε 0 so τ ν ad this cotradicts (i). It is clear that (ii) ad (iv) are equivalet, ad that (ii) implies (iii). It remais to show (ii) implies (i). If (iii) holds ad ν K, the bouded covergece theorem gives fdν = c(f) f C(X; R) sothatν is uiquely determied.

9 CONVERGENCE OF THE RUELLE OPERATOR 335 The above proof gives the followig more geeral theorem. Let X be a compact metric space ad for each 1letP : C(X; R) C(X; R) be liear, cotiuous, positive operators with P (1) = 1 ad P P m = P max(,m). Defie K = {ν M(X) P ν = ν} ad the K 1 K 2 ad K = 1 K is oempty. Oe ca show K has oly oe member iff f C(X; R) c(f) R with P f c(f). We wat to ivestigate the extreme poits of K, ad the followig lemma is helpful. Lemma Let ν K () ad let h: X [0, ) be measurable ad hdν =1. The h ν K () iff P () h = h a.e. (v). Proof. If h ν K (),the A B(X) hdν = χ A hdν = (P () χ A ) hdν by Corollary 2.5 applied to h ν A = P () (P () χ A h)dν by Corollary 2.5 applied to ν = P () χ A P () hdν by Theorem 2.1 (iii) & (vi) = P () (χ A P () h)dν by Theorem 2.1 (iii) & (vi) = χ A P () hdν by Corollary 2.5 applied to ν = P () hdν. A Hece P () h = h a.e. (v). Coversely, if P () h = h a.e. (v), the for f C(X; R), f hdν = P () (f h)dν by Corollary 2.5 applied to ν = P () f hdν by Theorem 2.1 (iii). Therefore h ν K (). This leads to Theorem (i) The extreme poits of K areexactlytheelemetsofk that are tail-trivial. (ii) If ν 1,ν 2 are extreme poits of K, the either ν 1 = ν 2 or ν 1,ν 2 are sigular o B = =0 T B(X). Proof. (i) Let ν be tail-trivial ad ν K.Letν = pν 1 +(1 p)ν 2 with 0 <p<1 ad ν 1,ν 2 K. Sice ν 1 ν ad ν 2 ν, the above lemma gives ν 1 = h 1 ν, ν 2 = h 2 ν with h 1,h 2 both measurable (mod ν) with respect to B. Therefore h 1,h 2 are both costat (mod ν) soν 1 = ν = ν 2. Hece ν is a extreme poit of K. If ν K ad ν is ot tail-trivial, there is some B 0 B with 0 <ν(b 0 ) < 1. The χb 0 ν ν(b, χ X\B0 ν 0) ν(x\b 0) K by Lemma 2.10, ad ν = ν(b 0 ) ( χb 0 ν ν(b )+ 0) (1 ν(b 0 ))( χ X\B 0 ν ν(x\b )sothatν is ot a extreme poit of K 0).

10 336 PETER WALTERS (ii) Let ν 1,ν 2 K ad both be tail-trivial. If ν 1 ν 2,choosef 0 C(X; R) ν 1 (f 0 ) ν 2 (f 0 ). Let B i = {x X (P f 0 )(x) ν i (f 0 )}. By the Martigale theorem ν i (B i ) = 1 ([P], p. 231). Sice B 1 B 2 = we have that ν 1,ν 2 are a sigular pair. Note that B i B. We wat to show that whe satisfies Bowe s coditio there are o sigular pairs i, ad hece, by Theorem 2.11 (ii), K as oly oe member. We shall use the ope Bowe balls B (x; δ) ={y X : d(t i x, T i y) <δ,0 i 1}. For M 1adε>0lets M (ε) deote the maximum umber of poits i a (M,ε) separated set with respect to T ([W2], p. 169). For the followig lemma we oly eed T to be positively expasive. Lemma If T : X X is positively expasive ad 2δ is a expasive costat, the for M 1, 0 ad y, z X the set B (y; δ) T (+M) z has at most s M (2δ) poits. Proof. If w 1,w 2 B (y; δ) T (+M) z,thed(t i w 1,T i w 2 ) < 2δ for 0 i 1 ad T +M w 1 = T +M w 2.Sice2δ is a expasive costat, the set T (B (y; δ) T (+M) z) has the same cardiality as B (y; δ) T (+m) z,aditmustbe(m,2δ) separated. Corollary With the otatio of Lemma 2.12 {v T (+M) z B (v; δ /2 ) B (y; δ /2 ) } has at most s M (2δ) poits. Proof. Let v be i the above set. The v B (y; δ) sov T (+m) z B (y; δ), ad we ca apply Lemma The followig theorem cocers strictly positive cotiuous fuctios o X, ad we ca write such a fuctio as f = e F for F C(X; R). Theorem Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio property. Let C(X; R). For sufficietly small δ>0 M 1 with the property that 1, x, z X, F C(X; R) (L +M where A = s M (2δ)e M +v(t,δ). e F )(x) e v(f,δ) A (L +M e F )(z) Proof. Let δ > 0besosmallthat2δ is a expasive costat. By weak specificatio there exist M 1withT +M B (w; δ /4 )=X 1, w X. Hece z,y X 1 B (y; δ /2 ) T (+M) z. Let F C(X; R) ad z,x X. The y T (+M) x )(y) e F (y) e(t+m e M (2δ) ν T (+M) z )(z)+m +v(t,δ) e F (v)+v(f,δ) by associatig to each e(t+m y T (+M) x those v T (+M) z with B (y; δ /2 ) B (v; δ/2). For each y this set of v s is oempty by choice of M above, ad has at most s M (2δ) members by Corollary This gives (L +M e F )(x) e v(f,δ) A (L +M e F )(z). Corollary Let T, be as i Theorem 2.14, ad for sufficietly small δ>0 let M be give by Theorem The 1, x, z X, F C(X; R) (P +M e F )(x) e v(f,δ) A 2 +M (P e F )(z), where A = s M (2δ)e M +v(t,δ).

11 CONVERGENCE OF THE RUELLE OPERATOR 337 Proof. By Theorem 2.14 we have (L +M ad by Theorem 2.14 with F =0wehave (L +M e F )(T +M x) e v(f,δ) A (L +M e F )(T +M z), 1)(T +M z) A (L +M 1)(T +M x). Theorem Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let C(X; R) satisfy lim if v (T, δ) < for some δ>0, which is implied by Bowe s coditio. The: (i) K has oly oe member which is the uique ν M(X) with L ν = ν for some >0. (ii) ν is tail-trivial. (iii) f C(X; R)( L f L 1 )(x) ν(f). (iv) The uique >0 determied by (i) satisfies = L 1 dν = e P (T,) = lim ( L +1 1 L 1 ) (x) x X. Also, is the spectral radius of L : C(X; R) C(X; R). (v) If satisfies Bowe s coditio, there is a costat D>1sothat D 1 L 1(x) D 0, x X. Proof. We first use Corollary 2.15 to show K has oly oe member. let ν, ν K. If we itegrate the iequality i Corollary 2.15 i x with respect to ν, ad itegrate i z with respect to ν we get e F dν e v(f,δ) A 2 e F dν 1, F C(X; R). If A = lim if A, which is fiite by assumptio, the e F dν A 2 e F dν f C(X; R). This implies ν ν, so by Theorem 2.11 (ii) we have ν = ν. To see the above iequality gives ν ν, let C be a closed subset of X with ν (C) = 0. For ε > 0chooseaopesetU ε C with ν (U ε ) < ε, ad a Urysoh fuctio f C(X; R) withχ C f χ Uε. The τ >0 (f + τ)dν A 2 (f + τ)dν so ν(c) +τ A 2 (ν (U ε )+τ) A 2 (ε + τ), ad hece ν(c) =0. IfB B(X) has ν (B) =0,theν (C) = 0 for every closed C B ad so ν(b) =sup{ν(c) C closed, C B} = 0. Hece K has oly oe member. From Corollary 2.3 we kow there exists ν M(X) ad>0withl ν = ν, ad that such a measure is i K. Hece there is a uique such ν. The is give by = L 1 dν. Theorem 2.11 (i) gives that ν is tail-trivial. The covergece property (iii) follows by Theorem 2.9. If we put f = L 1i the covergece property we get L+1 1 L 1 (x). Hece [(L 1)(x)] 1/ ad sice L = L 1,wegetthat is the spectra radius of L : C(X; R) C(X; R). The relatio P (T,)=log follows from 1 L 1(x) log ad Theorem 1.2. To prove (v) let δ be small eough for Theorem 2.14 to hold ad let M be give by Theorem Puttig F 0 i Theorem 2.14 gives L +M 1(x) A L +M 1(z) 1, x, z X

12 338 PETER WALTERS where A =supa. If we itegrate with ν i z we get L+M 1(x) X, 1, ad if we itegrate i x we get A 1 +M A x L+M 1(z) z X, 1. If we +M let B =sup{ Li 1(x) 0 i M, x X} ad E =if{ Li 1(x) i 0 i M, x X} > i 0, the put D =max(a,b,e 1 ). The D 1 L 1(x) D x X, 0. Corollary Let T be a positively expasive local homeomorphism satisfyig the weak specificatio coditio, ad let C(X; R) satisfy Bowe s coditio. If ν, are as i Theorem 2.16 the followig statemets are pairwise equivalet: (i) h C(X; R) with h>0, L h = h, ν(h) =1ad f C(X; R) (L f)(x) h(x)ν(f). (ii) h C(X; R) with h>0 ad L h = h. (iii) (L 1)(x) coverges uiformly as. (iv) 1 1 (L i 1)(x) i=0 coverges uiformly as. i (v) { L 1 0} is a equicotiuous subset of C(X; R). (vi) { 1 1 L i 1 i=0 1} is a equicotiuous subset of C(X; R). i Proof. Clearly (i) (ii). If (ii) holds, the puttig f = h i Theorem 2.16 (iii) gives (L 1)(x) h(x) ν(h), so (iii) holds. Clearly (iii) (iv), (iv) (vi), (iii) (v) ad 1 i=0 (v) (vi). It remais to show (vi) (i). If (vi) holds, the { 1 1} i is equicotiuous ad each member is bouded from below by D 1 ad above by D, by Theorem 2.16 (v). It s closure i C(X; R) is therefore compact so that L i 1 i there is a subsequece 1 j 1 j i=0 which coverges i C(X; R) tosomeh with D 1 h. The L h = h, adν(h) = 1. Puttig f = h i Theorem 2.16 (iii) gives (L 1)(x) h(x) ad hece Theorem 2.16 (iii) gives L f hν(f). Due to Theorem 2.16 (v), part (v) of Corollary 2.17 is equivalet to ε >0 δ >0 such that wheever d(x, x ) <δ,the (L 1)(x) (L 1)(x ) 1 <ε 1. We shall deduce uiqueess of the equilibrium state of a Bowe fuctio i 4 ad also cosider covergece of L f. Oe has the followig characterizatio of the Bowe coditio. Theorem Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio property. For C(X; R) the followig are pairwise equivalet: (i) satisfies Bowe s coditio. (ii) ν M(X) ad >0 with the property that for all sufficietly small δ> 0 D δ > 1 with D 1 δ ν(b(x;δ)) D e (T)(x) δ x X, 1. (iii) ν M(X) ad >0 with the property that for some expasive costat δ 0 D δ0 > 1 with D 1 δ 0 ν(b (x; δ 0 )) e D (T)(x) δ 0 x X, 1. Proof. To show that (ii) ad (iii) are equivalet it suffices to show that if δ 1,δ 2 are expasive costats ad the property holds for δ 1, the it holds for δ 2. So assume D δ1 exists. Choose N so that d(t i x, T i y) <δ 1,0 i N 1, implies d(x, y) <δ 2 L i 1

13 CONVERGENCE OF THE RUELLE OPERATOR 339 ad choose K so that d(t i x, T i y) <δ 2,0 i K 1, implies d(x, y) <δ 1.The B +N (x; δ 1 ) B (x; δ 2 )so ( ) N 1 D 1 e e δ 2 ν(b (x; δ 2 )) x X, 1, e (T)(x) ad B (x; δ 2 ) B K (x; δ 1 )for K so ν(b(x;δ2)) D e (T)(x) δ2 (e e ) K for K. If C =sup{ ν(b(x;δ2)) x X, 1 K}, the put e (T)(x) D δ2 =max(c, D δ1 (e e ) max(k,n) ). Hece (ii) ad (iii) are equivalet. Assume (i) holds ad let δ>0 be so small that it is a expasive costat ad C =sup 1 v (T, δ) <. Let ν M(X) ad>0besothatl ν = ν. By Corollary 2.8 L fdν = fdν for f L(X; R)withf 0. By weak specificatio there is M 1withT +M B (x; δ) =X x X, 1. The L +M χ B(x;δ)(z) = e (T+M )(y) χ B(x;δ)(y) y T (+M) z has ozero terms for each z X ad e (T)(x) M C L +M χ B(x;δ)(z) s M (2δ)e T(x)+M +C by Lemma Itegratig i z gives e (T)(x) M C +M ν(b (x; δ)) e T(x) s M (2δ)e M +C x X, 1. Hece (ii) holds with D δ = s M (2δ)e M +C M. Now suppose (ii) holds ad we wish to prove (i). For sufficietly small δ ν(b(x;2δ)) ν(b (x;δ)) D 2δ D δ x X, 1. Let d(t i x, T i x ) δ /2 for 0 i 1. The x B (x ; δ) sob (x; δ) B (x ;2δ) ad e T(x) T(x ) Dδ 2 ν(b (x; δ)) ν(b (x ; δ)) D2 δ Therefore satisfies Bowe s coditio. ν(b (x ;2δ)) ν(b (x ; δ)) D 2δD 3 δ. 3. g-measures Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let g : X (0, 1) be cotiuous ad satisfy y T 1 x g(y) =1 x X. Such a fuctio is called a g-fuctio for T,adassigs, i a cotiuous way, a probability distributio to each of the fiite sets T 1 x. We have (L log g f)(x) = y T 1 x g(y)f(y) adl log g1 =1sothat(P () log g f)(x) = z T T x g(z) g(t 1 z)f(z). Hece P () log g = U T L() log g. By the Schauder-Tychaoff fixed poit theorem there is always at least oe µ M(X) withl log g µ = µ. Everyµ M(X) withl log g µ = µ is T -ivariat, sice if f C(X; R) f Tdµ= L log g (f T ) dµ = fdµ.ifact, usig the otatio of the previous sectio, we have, if G g = {µ M(X) L log g µ = µ} is the set of g-measures: Theorem 3.1. G g = K log g M(X, T)=K (1) log g M(X, T). Proof. We have G g K log g by Corollary 2.3 ad G g M(X, T) bytheabove. Hece G g K log g M(X, T) K (1) log g M(X, T). However, if µ K(1) log g M(X, T) the µ G g.

14 340 PETER WALTERS From Theorem 2.9 we kow that K log g has oly oe member iff f C(X; R) c(f) R with L log g f c(f). Oe ca easily show that there is a uique g-measure iff f C(X; R) c(f) R with 1 1 i=0 Li log g f c(f). Bramso ad Kalikow have give examples of g-fuctios without uique g-measures ([B-K1]). Our Theorem 2.16 gives the followig result, which was proved by Fa i the case of topologically mixig subshifts of fiite type ([F]). Theorem 3.2. Let T : X X be a positively expasive local homeomorphism with the weak specificatio coditio. Let g : X (0, 1) be cotiuous ad x X y T 1 x g(y) =1. Let log g satisfy Bowe s coditio (i.e. δ >0 ad C>0 such that d(t i x, T i x g(x)g(tx) g(t ) δ, 0 i 1 implies 1 x) g(x )g(tx ) g(t 1 x ) e C ). The f C(X; R) L log g f costat. The set K log g has oly oe member, i particular, there is a uique g-measure µ. Also, µ is exact. So for the collectio of fuctios of the form log(g) Bowe s assumptio implies the coclusio of Ruelle s theorem. There are examples of g s with L log g f costat but log g does ot satisfy Bowe s coditio ([H]). Such a example is the followig where a (0, 1). O the space X = {0, 1} Z+ let ( 3 +1 a +2) if (x 0,...,x +1 )=(1, 1,...,1, 1, 0), 0, ( 3 +1 g(x) = 1 a +2) if (x 0,...,x +1 )=(0, 1, 1,...,1, 1, 0), 0, a if x i =1 i 0, 1 a if x 0 =0adx i =1 i 1. Theorem 2.18 gives the followig Theorem 3.3. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. For a g-fuctio g : X (0, 1) the followig are equivalet: (i) log g satisfies Bowe s coditio. (ii) µ M(X, T) with the property that for every sufficietly small δ>0 D δ > 1 with D 1 δ µ(b (x; δ)) g(x) g(t 1 x) D δ x X, 1. (iii) µ M(X, T) such that for some expasive costats δ 0 D δ0 > 1 with D 1 δ 0 µ(b (x; δ 0 )) g(x) g(t 1 x) D δ 0 x X, 0. Proof. I the proof of Theorem 2.18 we showed a measure satisfies the property i (ii) iff it satisfies the property i (iii). If (i) holds ad we choose µ with L log g µ = µ, the the proof of Theorem 2.18, with ν = µ, =1,showsµ satisfies (ii). We get (iii) implies (i) by Theorem Corollary 4.7 will give more iformatio.

15 CONVERGENCE OF THE RUELLE OPERATOR Equilibrium states I this sectio we wat to study equilibrium states ad study the covergece of L f as. We use a method ivolvig measurable g-fuctios. Let T : X X be a positively expasive local homeomorphism with the weak specificatio property. If C(X; R) satisfies Bowe s coditio, Theorem 2.16 gives a uique ν M(X) ad uique >0withL ν = ν, equals ep (T,) ad ν is tail-trivial. Also D >1withD 1 L 1(x) D x X, 1. The followig gives iformatio about solvig the equatio L h = h whe C(X; R). Theorem 4.1. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. For C(X; R) ad >0 the followig statemets are equivalet: (i) D >1 with D 1 L 1(x) D x X, 1. (ii) measurable h: X [d 1,d 2 ] (0, ) with L h = h. (iii) h: X [d 1,d 2 ] (0, ) with L h = h. L Proof. Assume (i) holds. Let h 1 (x) = lim sup 1(x).Theh 1 : X [D 1,D] is measurable ad oe readily gets h 1 (x) (L h 1 )(x). To get h with L h = h everywhere we ca proceed as follows. We have (L h1) D L 1(x) D 2 1, x X ad sice D 1 h 1 L h 1 L2 h1 (L..., if we let h(x) = lim h1)(x) (L 2 =sup h1)(x),theh is measurable, h: X [D 1,D 2 ], ad L h = h everywhere. Hece (ii) holds. Clearly (ii) implies (iii). It remais to show (iii) implies (i). From (L h)(x) = h(x) wehaved 1 L 1(x) d 2 so that L 1(x) d2 d 1. Similarly, d 2 L 1(x) d 1 so d1 d 2 L 1(x). The discussio before Theorem 4.1 gives Corollary 4.2. If satisfies Bowe s coditio ad = e P (T,),thereexistsa measurable h: X [d 1,d 2 ] (0, ) with L h = h everywhere. From ow o we assume C(X; R) satisfies Bowe s coditio ad that ν M(X), >0, h: X [d 1,d 2 ] (0, ) aresothatl ν = ν, h is measurable, L h = h ad ν(h) = 1. The last coditio ca be attaied by replacig h i Corollary 4.2 by h ν(h). If we wat to emphasize depedece o we use ν,,h. Theorem 4.3. Let satisfy Bowe s coditio ad let ν,, h be as above. Let µ = h ν. Theµ M(X, T) ad T is a exact edomorphism with respect to µ. If h : X (0, ) is measurable with ν(h )=1ad L h = h a.e. (ν), theh = h a.e. (ν). Proof. If f C(X; R), the f T hdν = 1 L (f T h)dν = 1 f L hdν = f hdν,soµ is T -ivariat. The measures ν ad h ν are equivalet so µ is also tail-trivial, ad hece T is exact with respect to µ. If h is as i the statemet of the theorem, the µ = h ν is also i M(X, T) ad is tail-trivial ad so both µ ad µ are ergodic. Sice µ µ we must have µ = µ ad hece h = h a.e. (ν). From ow o we use µ for h ν, adwriteitµ if we wish to emphasize its relatioship to. Note L p ν(x) = L p µ(x) p 1. Also, let g = e h h T. The

16 342 PETER WALTERS g is measurable, g : X [a, b] (0, 1) ad x X y T 1 x g(y) = 1. Let L log g : L(X; R) L(X; R) be defied by (L log g f)(x) = y T 1 x g(y)f(y). We have for 1 (L 1 log gf)(x) = h(x) (L (h f))(x). From Corollary 2.8 L fdν = fdν f L 1 ν(x) sowehave L log g fdµ = fdµ f L 1 ν(x) =L 1 µ(x). We have the followig versio of a result of Ledrappier ([L]). We write B istead of B(X) adifτ M(X, T), the h τ (T ) deotes the etropy of the measurepreservig trasformatio T :(X, B,τ) (X, B,τ). Lemma 4.4. Let T : X X be a positively expasive cotiuous surjectio ad let g : X [a, b] (0, 1) be measurable ad satisfy y T 1 x g(y) =1 x X. We have h τ (T )+ log gdτ 0 τ M(X, T). Forσ M(X, T) the followig statemets are pairwise equivalet: (i) L log g fdσ = fdσ f L 1 σ(x). (ii) σ M(X, T) ad f L 1 σ(x) E σ (f /T 1 B)(x) = y T 1 Txg(y)f(y) a.e. (σ). (iii) σ M(X, T) ad h σ (T )+ log gdσ=0. Proof. We write L istead of L log g. Sice T is positively expasive, every fiite partitio of X ito sets of sufficietly small diameter is a oe-sided geerator. Hece h τ (T )=H τ (B /T 1 B) τ M(X, T). If g τ : X [0, 1] is defied a.e. (τ) by E τ (f /T 1 B)(x) = g τ (y)f(y), f L 1 τ, the Hece y T 1 Tx h τ (T )=H τ (B /T 1 B)= = h τ (T )+ log g τ dτ. y T 1 Tx log gdτ = log g/g τ dτ ( ) g 1 dτ g τ = g τ (y) y T 1 Tx =0sice y T 1 Tx g τ (y)logg τ (y) dτ(x) ( ) g(y) g τ (h) 1 dτ(x) g(y) =1 x X. Equality holds here iff log g/g τ = g/g τ 1a.e.(τ) i.e. g = g τ a.e. (τ). Hece h τ (T )+ log gdτ 0 τ M(X, T).

17 CONVERGENCE OF THE RUELLE OPERATOR 343 (i) (ii) Let f L 1 σ.the f Tdσ= L(f T ) dσ = fdσ so σ M(X, T). Also, if B B fdσ = f χ B Tdσ= L(f χ B T ) dσ T 1 B = (L(f χ B T )) Tdσ= g(y)f(y) dσ(x). T 1 B y T 1 Tx (ii) (iii) By the above if (ii) holds, the g = g σ ad (iii) holds. (iii) (i) From the above proof we kow that for τ M(X, T) h τ (T )+ log gdτ =0iffg τ = g a.e. (τ). So if (iii) holds, the g σ = g a.e. (σ). Hece for f L 1 σ Lfdσ = (Lf) Tdσ= g(y)f(y) dσ(x) = fdσ. y T 1 Tx Sice our measure µ = h ν satisfies statemet (i) of Lemma 4.4, it also satisfies (ii) ad (iii). We use this to show µ is the uique equilibrium state of. Recall that µ is a equilibrium state of if τ M(X, T) h τ (T )+ dτ h µ (T )+ dµ; equivaletly if h µ (T )+ dµ= P (T,). Theorem 4.5. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let C(X; R) satisfy Bowe s coditio. The has a uique equilibrium state ad this state is µ = h ν. With respect to µt is exact. Proof. Sice log = P (T,), we kow σ M(X, T) is a equilibrium state for iff h σ (T )+ dσ =log. Sice =logg +log +logh T log h this is equivalet to h σ (T )+ log gdσ = 0. By Lemma 4.4 this is equivalet to L log g fdσ = fdσ f L 1 σ (X), which is equivalet to L fdτ = fdτ f L 1 τ(x), where τ = 1 h σ. This gives L τ = τ ad we kow, by Theorem 2.16 (i), that ν is the oly probability measure that satisfies this. Hece µ = h ν is a equilibrium state for. If σ is a equilibrium state for, theσ = c h ν for some c>0. Sice σ ad h ν are both probability measures we get c =1,adµ = h ν is the oly equilibrium state for. Exactess was proved i Theorem 4.3. Theorem 4.6. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let C(X; R) satisfy Bowe s coditio, ad let = e P (T,). The followig three statemets about σ M(X, T) are pairwise equivalet: (i) σ is the uique equilibrium state µ = h ν. (ii) For every sufficietly small δ>0 E δ > 1 with E 1 δ σ(b (x; δ)) e (T)(x) E δ x X, 1. (iii) For some expasive costat δ 0 E δ0 > 1 with E 1 δ 0 σ(b (x; δ 0 )) e (T)(x) E δ 0 x X, 1.

18 344 PETER WALTERS Proof. Let ν,, h be associated to as usual. Assume (i). We have 0 < d 1 h d for some d > 1. By the proof of Theorem 2.18 we kow, ν satisfy statemet (ii) of Theorem 2.18 so we have E 1 δ µ(b (x; δ)) e (T)(x) E δ x X, 1 if E δ = dd δ. Hece (ii) holds. Clearly (ii) implies (iii). Assume (iii) holds ad we prove (i). We have 1 log σ(b (x; δ 0 )) + 1 (T )(x) log = P (T,). Itegratig with respect to σ, ad usig the Bri-Katok local etropy formula ([B-K2]), gives h σ (T )+ dσ = P (T,). This says σ is a equilibrium state for ad so must equal the uique oe µ. We have the followig special case. Corollary 4.7. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let g C(X; R) be a g-fuctio so that log g satisfies Bowe s coditio. The the uique g-measure is the oly member µ of M(X, T) with either of the followig properties: (i) for every sufficietly small δ>0 E δ > 1 with E 1 δ µ(b (x; δ)) g(x)g(tx) g(t 1 x) E δ x X, 1; (ii) for some expasive costat δ 0 E δ0 > 1 with E 1 δ 0 µ(b (x; δ 0 )) g(x)g(tx) g(t 1 x) E δ 0 x X, 1. The followig geeralizes Theorem 3.3 Theorem 4.8. Let T : X X be a positively expasive local homeomorphism with the weak specificatio coditio. The followig statemets about C(X; R) are pairwise equivalet: (i) satisfies Bowe s coditio. (ii) µ M(X, T) ad >0 with the property that for all sufficietly small δ>0 E δ > 1 with E 1 δ µ(b (x; δ)) e (T)(x) E δ x X, 1. (iii) µ M(X, T) ad >0 with the property that for some expasive costat δ 0 E δ0 > 1 with E 1 δ 0 µ(b (x; δ 0 )) e E (T)(x) δ 0 x X, 1. Proof. We get (i) (ii) by Theorem 4.6. Clearly (i) (iii), ad we get (iii) (i) by Theorem 2.8. We ca ow make some deductios about equilibrium states usig a result of Quas [Q]. The result of Quas says, i our cotext, that if T : X X is a positively expasive local homeomorphism satisfyig the weak specificatio property ad µ is a ergodic member of M(X, T) with full support, the if F L µ (X) satisfies F T F = f C(X; R) a.e. there is G C(X; R) withg T G = f everywhere.

19 CONVERGENCE OF THE RUELLE OPERATOR 345 We shall let C(X; R) satisfy Bowe s coditio ad let, ν, h be so that L ν = ν, L h = h, ν(h) = 1. We kow, by Theorem 2.8, that ν(b (x, δ)) > 0 x X, 1sothatifU is ope, ν(u) > 0 because B (x, δ) U for some 1 ad some x. Hece µ = h ν is a ergodic T -ivariat measure with full support. Theorem 4.9. Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditios. Let 1, 2 C(X; R) both satisfy Bowe s coditio. The µ 1 = µ 2 iff c R ad f C(X; R) with 1 2 = c+f T f. Proof. Let µ 1 = µ 2 = µ. The g 1 = g 2 a.e. µ so e 1 h 1 = e 2 h 2 1h 1 T 2h 2 T a.e. (µ) where the i,h i correspod to i, i =1, 2. Hece 1 2 = log( 1 / 2 )+H T H a.e. (µ) whereh =logh 1 log h 2.SiceH L µ (X), the result of Quas metioed above implies f C(X; R) with 1 2 = log( 1 / 2 )+f T f everywhere. Coversely, if 1 2 = c + f T f with f C(X; R), the 1, 2 have the same equilibrium states, sice σ( 1 )=σ( 2 )+c σ M(X, T), so µ 1 = µ 2 by Theorem 4.5. This result was kow uder the stroger assumptio of Theorem 1.3 ([W1], p. 134). Theorem Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let C(X; R) satisfy Bowe s coditio. The equivalet statemets (i) to (vi) of Corollary 2.17 are each equivalet to: (vii) the uique equilibrium state µ of is a g-measure for a cotiuous g : X (0, 1). Proof. we show (vii) is equivalet to the existece of h C(X; R) withh>0 ad L h = h, which is statemet (ii) of Corollary If such a h exists, the g = is cotiuous ad µ is a g-measure for this g, so that (vii) holds. Now assume (vii) holds. If µ is a g-measure for a cotiuous g, theg = e h h T a.e. (µ) where L h = h ad h: X [d 1,d 2 ] (0, ) is measurable. This says log g log =logh T log h a.e. (µ), ad sice the left-had side is cotiuous ad log h L µ (X) wecausequas resulttoobtaih C(X; R) with log g log = H T H everywhere. Sice y T 1 x g(y) =1 x X we have L e H = e H. e h h T We ow tur to the covergece of L f. I the followig we use p to deote the orm i L p ν(x). The followig result was obtaied by Ruelle by aother method ([Ru]). Theorem Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. Let C(X; R) satisfy Bowe s coditio. Let >0 ad v M(X) satisfy L ν = ν ad let h: X [d 1,d 2 ] (0, ) be measurable with L h = h ad ν(h) =1. For every p 1 ad every f L p ν(x) (L f)(x) h(x)ν(f) 0 as. p Proof. L f(x) h(x)ν(f) =[L log g (f/h)(x) ν(f)]h(x) =[L log g(f/h)(x) µ(f/h)]h(x).

20 346 PETER WALTERS Sice 0 <d 1 h d 2,wehavef L p ν (X) ifff/h Lp µ (X), ad it suffices to show (L log g F )(x) µ(f ) p 0 F L p µ(x), where p ow deotes the orm i L p µ (X). By the L1 ad L p, p>1, covergece theorems ([P], pp. 231, 234) (X). Sice µ is tail-trivial, the limit is µ(f ). Hece (L log g F )(T x) µ(f ) p 0so,siceµ is T -ivariat, (L log g F )(x) µ(f ) p 0. E µ (F/T B(X)) E µ (F/ =0 T B(X)) i L p µ Corollary Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio coditio. There is a icreasig sequece of itegers 1 < 2 < 3 adasetb B(X) with µ(b) =1,whereµ is the uique equilibrium state of, such that f C(X; R) x B (L i f)(x) h(x)ν(f). i Proof. By Theorem 4.11 with f =1weget{ i } ad B B(X) withν(b) =1so that (L i 1)(x) i result. h(x) x B. Combiig this with Theorem 2.16 (iii) gives the 5. Properties of the desity fuctio h Let T : X X be a positively expasive local homeomorphism satisfyig the weak specificatio property, ad let C(x; R) satisfy Bowe s coditio. From Corollary 4.2 we kow there is a measurable h: X [d 1,d 2 ] (0, ) withl h = h ad we ca ormalize so that ν(h) =1. Let h: X [d 1,d 2 ] be defied by h(x) = lim δ 0 sup{h(y) d(x, y) <δ}. Theh is upper semi-cotiuous ad h L h. Sice both sides have the same itegral with respect to ν, we get h = L h a.e. (ν). Similarly, if h: X [d 1,d 2 ]is defied by h(x) = lim δ 0 if{h(y) d(x, y) <δ}, theh is lower semi-cotiuous ad h L h so h = L h a.e. (ν). We have h h h so that ν(h) 1 ν(h), ad by Theorem 4.3 h = a.e. (ν). h ν(h) = h h ν(h) h ν(h) Theorem 5.1. We have ν(h) everywhere. Also, { x X h(x) ν(h) = h(x) } {x X h ad h are cotiuous at x} ν(h) ad both sets are dese G δ sets with full ν-measure. Proof. Let c = ν(h) ν(h) 1. The h = ch is upper semicotiuous so that {x (h = ch)(x) < 0} is ope ad has ν-measure zero. Hece this set is empty ad ch h everywhere. We have {x X ch(x) =h(x)} = =1 { x X (h ch)(x) < 1 } so this set is a G δ set, ad sice it has ν-measure 1 it must be dese. Let x be so that ch(x) =h(x) ad we wat to show h ad h are cotiuous at x. Let ε>0. Sice h is upper semicotiuous at x ad h is lower semicotiuous at x δ >0 so that d(z,x) <δimplies h(z) < h(x) +ε ad h(x) ε<h(z). Therefore if d(z,x) <δ,the h(x) cε = c(h(x) ε) <ch(z) h(z) < h(x)+ε

21 CONVERGENCE OF THE RUELLE OPERATOR 347 so h(z) h(x) <cε. Also, d(z,x) <δimplies ( h(z) = 1+ 1 ) ε< 1c ( c h(z) 1+ 1 ) ε< 1 h(x) ε = h(x) ε<h(z) c c so that h(z) h(x) <ε. h(x) Corollary 5.2. If if x h(x) =1(i particular, if h is cotiuous at oe poit), the ν(h) =ν(h) ad ν({x X h is cotiuous at x}) =1. Proof. We have 1 ν(h) ν(h) h(x) h(x) x X so the assumptio implies ν(h) =ν(h). By Theorem 5.1 {x X h(x) =h(x)} {x X h ad h are cotiuous at x} so if h(x) =h(x), the usig h h h gives h(z) h(x) h(z) h(x) h(z) h(x) ad hece h is cotiuous at x. Therefore the desity dµ dν has a versio h ν(h) cotiuous at ν-almost every poit, ad a versio which is upper semicotiuous ad is h ν(h) which is lower semicotiuous ad is cotiuous at ν-almost every poit. The author does ot kow if there exists a cotiuous desity h whe satisfies Bowe s coditio. Refereces [A-H] N. Aoku ad K. Hiraide, Topological Theory of Dyamical Systems, North-Hollad, MR 95m:58095 [B1] R. Bowe, Some systems with uique equilibrium states, Math. Systems Theory 8 (1974), MR 53:3257 [B2] R. Bowe, Equilibrium States ad the Ergodic Theory of Aosov Diffeomorphisms, Lecture Notes i Math., vol. 470, Spriger, Berli, MR 56:1364 [B-K1] M. Bramso ad S. Kalikow, Nouiqueess i g-fuctios, IsraelJ.Math.84 (1993), MR 94h:28011 [B-K2] M. Bri ad A. Katok, O local etropy, igeometric Dyamics, Lecture Notes i Math., Vol. 1007, Spriger, Berli, MR 85c:58063 [F] Ai Hua Fa, A proof of the Ruelle operator theorem, Rev. Math. Phys. 7 (1995), MR 97e:28034 [H] F. Hofbauer, Examples for the ouiqueess of the equilibrium state, Tras. Amer. Math. Soc. 228 (1977), MR 55:8312 [L] F. Ledrappier, Pricipe variatioel et systèmes dyamiques symboliques, Z. Wahr. ud Verw. Gebiete 30 (1974), MR 53:8384 [P] K. Parthasarathy, Itroductio to Probability ad Measure, Macmilla, Lodo, MR 58:31322a [Q] A. Quas, Rigidity of cotiuous coboudaries, Bull. Lodo Math. Soc. 29 (1997), MR 99c:28054 [Re] W. Reddy, Expadig maps o compact metric spaces, Topology Appl. 13 (1982), MR 83d:54070 [Ru] D. Ruelle, Thermodyamic formalism for maps satisfyig positive expasiveess ad specificatio, Noliearity 5 (1992), MR 94a:58115 [W1] P. Walters, Ivariat measures ad equilibrium states for some mappigs which expad distaces, Tras. Amer. Math. Soc. 236 (1978), MR 57:6371 [W2] P. Walters, A Itroductio to Ergodic Theory, Graduate Texts i Math., vol. 79, Spriger, Berli, MR 84e:28017 Uiversity of Warwick, Mathematics Istitute, Covetry CV4 7AL, Eglad address: pw@maths.warwick.ac.uk