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

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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.

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.

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.

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