Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations
Harrison Bray, Richard Canary, Lien-Yung Kao, Giuseppe Martone
aa r X i v : . [ m a t h . D S ] F e b COUNTING, EQUIDISTRIBUTION AND ENTROPY GAPS AT INFINITYWITH APPLICATIONS TO CUSPED HITCHIN REPRESENTATIONS
HARRISON BRAY, RICHARD CANARY, LIEN-YUNG KAO, AND GIUSEPPE MARTONE
Abstract.
We show that if an eventually positive, non-arithmetic, locally H¨older continuous po-tential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity,then one may apply the renewal theorem of Kesseb¨ohmer and Kombrink to obtain counting andequidistribution results. We apply these general results to obtain counting and equidistributionresults for cusped Hitchin representations, and more generally for cusped Anosov representationsof geometrically finite Fuchsian groups. Introduction
In this paper, we use the Renewal Theorem of Kesseb¨ohmer and Kombrink [30] to establishcounting and equidistribution results for well-behaved potentials on topologically mixing countableMarkov shifts with (BIP) in the spirit of Lalley’s work [33] on finite Markov shifts. Inspired bywork of Schapira-Tapie [60, 61], Dal’bo-Otal-Peign´e [17], Iommi-Riquelme-Velozo [24] and Velozo[65] in the setting of geodesic flows on negatively curved Riemannian manifolds, we define notions ofentropy gap at infinity for our potentials. Our results require that the potentials are non-arithmetic,eventually positive and have an entropy gap at infinity.Our main motivation for this general analysis was provided by cusped Hitchin representationsof a geometrically finite Fuchsian group into SL ( d, R ). Given a linear functional φ on the Cartanalgebra a of SL ( d, R ) which is a positive linear combination of simple roots, we can define the φ -translation length ℓ φ ( A ) = φ ( ℓ ( A )) (where ℓ is the Jordan projection) for A ∈ SL ( d, R ). The firstconsequence of the general theory we develop is that if ρ is cusped Hitchin, then (cid:8) [ γ ] ∈ [Γ] | < ℓ φ ( ρ ( γ )) ≤ t (cid:9) ∼ e tδ tδ where δ = δ φ ( ρ ) is the φ -entropy of ρ (and [Γ] is the collection of conjugacy classes of elementsof Γ.) We also obtain a Manhattan curve theorem and equidistribution results in this context.In later work, we plan to use these results to construct pressure metrics on cusped Hitchin com-ponents. A longer term goal is the development of a geometric theory of the augmented Hitchincomponent which parallels the study of the augmented Teichm¨uller space as the metric completionof Teichm¨uller space with the Weil-Petersson metric (see Masur [41]). General Thermodynamical results:
We now give more precise statements of our generalresults. We assume throughout that (Σ + , σ ) is a topologically mixing, one-sided, countable Markovshift with alphabet A which has (BIP). Moreover, all of our functions will be assumed to be locallyH¨older continuous (see Section 2 for precise definitions).We now introduce the crucial assumptions we will make in our work. Given a locally H¨oldercontinuous function f : Σ + → R and a ∈ A , we let I ( f, a ) = inf (cid:8) f ( x ) | x ∈ Σ + , x = a (cid:9) and S ( f, a ) = sup (cid:8) f ( x ) | x ∈ Σ + , x = a (cid:9) . Note that I ( f, a ) and S ( f, a ) are finite since f is locally H¨older continuous. Canary was partially supported by grant DMS-1906441 from the National Science Foundation and grant 674990from the Simons Foundation.
We say that f has a strong entropy gap at infinity if the series Z ( f, s ) = X a ∈A e − sS ( f,a ) has a finite critical exponent d ( f ) > s = d ( f ).We say that f has a weak entropy gap at infinity if Z ( f, s ) has a finite critcial exponent d ( f ) > δ = δ ( f ) > d ( f ) > P ( − δf ) = 0 where P is the Gurevich pressure functionassociated to (Σ + , σ ) (defined in Section 2). We will see later (in Section 3), that a strong entropygap at infinity implies a weak entropy gap at infinity.We say that f is strictly positive if c ( f ) = inf { f ( x ) | x ∈ Σ + } >
0. We say that f is eventuallypositive if there exist N ∈ N and B > S n f ( x ) = f ( x ) + f ( σ ( x )) + · · · + f ( σ n − ( x )) > B for all n ≥ N and x ∈ Σ + . Recall that f is arithmetic if the subgroup of R generated by { S n f ( x ) | x ∈ Fix n , n ∈ N } is cyclic, where x ∈ Fix n if σ n ( x ) = x .We begin by stating our general counting results. For all n ∈ N , let M f ( n, t ) = { x ∈ Σ + : x ∈ Fix n and S n f ( x ) ≤ t } and let M f ( t ) = ∞ X n =1 n M f ( n, t ) . Theorem A (Growth rate of closed orbits) . Suppose that (Σ + , σ ) is a topologically mixing, one-sided, countable Markov shift which has (BIP). If f : Σ + → R is locally H¨older continuous, non-arithmetic, eventually positive and has a weak entropy gap at infinity, and P ( − δf ) = 0 , then lim t →∞ M f ( t ) tδe tδ = 1 . Similarly, for all k ∈ N , let R f ( k, t ) = { x ∈ M f ( k, t ) | x / ∈ M f ( n, t ) if n < k } and let R f ( t ) = ∞ X k =1 k R f ( k, t ) . If x ∈ M f ( n, t ) − R f ( n, t ), then there exists j ≥ x ∈ M f ( nj , tj ), so M f ( t ) − M f (cid:18) t (cid:19) ≤ R f ( t ) ≤ M f ( t ) . Therefore, the following result is an immediate corollary of Theorem A.
Corollary 1.1 (Growth rate of closed prime orbits) . Suppose that (Σ + , σ ) is a topologically mixing,one-sided, countable Markov shift which has (BIP). If f : Σ + → R is locally H¨older continuous,non-arithmetic, eventually positive and has a weak entropy gap at infinity, and P ( − δf ) = 0 , then lim t →∞ R f ( t ) tδe tδ = 1 . Another immediate corollary of our counting result is Bowen’s formula for the critical exponent.Let Σ f be the suspension flow of f and let O f be the collection of closed orbits of Σ f O f ( t ) = { λ | ℓ f ( λ ) ≤ t } where ℓ f ( λ ) is the period of λ . Notice that O f ( t ) = M f ( t ), since if λ ∈ O f ( t ), then there exists x ∈ Fix n for some n , so that S n f ( x ) = ℓ f ( λ ) and x is well-defined up to cyclic permutation. Corollary 1.2 (Bowen’s formula) . Suppose that (Σ + , σ ) is a topologically mixing, one-sided, count-able Markov shift which has (BIP). If f : Σ + → R is locally H¨older continuous, non-arithmetic,eventually positive, has a weak entropy gap at infinity and P ( − δf ) = 0 , then δ = lim t →∞ t log O f ( t ) . OUNTING AND EQUIDISTRIBUTION 3 If f : Σ + → R and g : Σ + → R are two eventually positive locally H¨older continuous functions,then there is a natural identification of the set O f of closed orbits of Σ f and the set O g of closedorbits of Σ g . If f is strictly positive and has a weak entropy gap at infinity so that P ( − δf ) = 0,then the equilibrium state for − δf induces a measure of maximal entropy on the suspension flowon Σ f . We obtain an equidistribution result for this equilibrium state which roughly says that itbehaves like a Patterson-Sullivan measure.In the following theorem, if φ and ψ are real-valued functions, we say that φ ∼ ψ if lim t →∞ φ ( t ) ψ ( t ) = 1 . Theorem B (Equidistribution) . Suppose that (Σ + , σ ) is a topologically mixing, one-sided, count-able Markov shift which has (BIP) and f : Σ + → R is locally H¨older continuous, non-arithmetic,eventually positive, has a weak entropy gap at infinity, P ( − δf ) = 0 and µ − δf is the equilibriumstate for − δf . If g : Σ + → R is locally H¨older continuous, eventually positive, and there exists C > such that | f ( x ) − g ( x ) | < C for all x ∈ Σ + , then X γ ∈O f ( t ) l g ( γ ) l f ( γ ) ∼ (cid:18) R g dµ − δf R f dµ − δf (cid:19) · e tδ tδ as t → ∞ . Equivalently, ∞ X k =1 k X x ∈M f ( k,t ) S k g ( x ) S k f ( x ) ∼ (cid:18) R g dµ − δf R f dµ − δf (cid:19) · e tδ tδ as t → ∞ . We can obtain a completely analogous statement if we instead consider the set P f of primitiveclosed orbits of the suspension flow Σ f .Suppose that f : Σ + → R is locally H¨older continuous, eventually positive, and has a strongentropy gap at infinity and that g : Σ + → R is also eventually positive and locally H¨older continuous,and that there exists C > | f ( x ) − g ( x ) | < C for all x ∈ Σ + . (Notice that this implies that d ( f ) = d ( g ).) Inspired by Burger [12], we define, the Manhattan curve C ( f, g ) = { ( a, b ) ∈ R | P ( − af − bg ) = 0 a ≥ , b ≥ , a + b > } . The Manhattan curve has the following properties.
Theorem C (Manhattan curve) . Suppose that (Σ + , σ ) is a topologically mixing, one-sided count-able Markov shift with (BIP), f : Σ + → R is locally H¨older continuous, eventually positive and hasa strong entropy gap at infinity and that g : Σ + → R is also eventually positive and locally H¨oldercontinuous. If there exists C > so that | f ( x ) − g ( x ) | < C for all x ∈ Σ + , then (1) ( δ ( f ) , , (0 , δ ( g )) ∈ C ( f, g ) . (2) If a ≥ , b ≥ , and a + b > , then there exists a unique t > d ( f ) a + b so that ( ta, tb ) ∈ C ( f, g ) . (3) C ( f, g ) is a closed subsegment of an analytic curve. (4) C ( f, g ) is strictly convex, unless S n f ( x ) = δ ( g ) δ ( f ) S n g ( x ) for all x ∈ Fix n and n ∈ N . OUNTING AND EQUIDISTRIBUTION 4
Moreover, the tangent line to C ( f, g ) at ( a, b ) ∈ C ( f, g ) has slope s ( a, b ) = − R Σ + g dµ − af − bg R Σ + f dµ − af − bg where µ − af − bg is the equilibrium state of the function − af − bg . Applications to cusped Hitchin representations:
Let S = H / Γ be a non-compact, geometri-cally finite, hyperbolic surface, and let Λ(Γ) ⊂ ∂ H be the limit set of Γ ⊂ H . Following Fock andGoncharov [20], a cusped Hitchin representation is a representation ρ : Γ → SL ( d, R ) such that if γ ∈ Γ is parabolic, then ρ ( γ ) is a unipotent element with a single Jordan block and there exists a ρ -equivariant positive map ξ ρ : Λ(Γ) → F d . If Γ is convex cocompact, cusped Hitchin representationsare simply the Hitchin representations studied by Labourie-McShane [32]. As these are covered bythe traditional theory of Anosov representations, we will focus on the case where Γ is not convexcocompact. If d = 3 and S has finite area, then a cusped Hitchin representation is simply theholonomy map of a finite area strictly convex projective structure on S (see Marquis [39]). Moregenerally, if ρ : Γ → SL ( , R ) acts geometrically finitely, in the sense of Crampon-Marquis [16, Def.5.14], on a strictly convex domain with C boundary, then ρ is cusped Hitchin by [20, 1.3. Thm.].Let a = { ~a ∈ R d | a + · · · + a d = 0 } be the standard Cartan algebra for the Lie algebra sl ( d, R ) of SL ( d, R ) . If T ∈ SL ( d, R ), let λ ( T ) ≥ · · · ≥ λ d ( T )be the (ordered) moduli of (generalized) eigenvalues of T (with multiplicity). The Jordan (orLyapunov) projection ℓ : SL ( d, R ) → a is given by ℓ ( T ) = (log λ ( T ) , · · · , log λ d ( T )) . For each k = 1 , . . . , d −
1, let α k : a → R be given by α k ( ~a ) = a k − a k +1 and let∆ = ( d − X k =1 t t α k | t k ≥ ∀ k and t k > k ) ⊂ a ∗ . For example, if α H is the Hilbert length functional given by α H ( ~a ) = a − a d , then α H = P d − k =1 α k ∈ ∆. Similarly, if ω ( ~a ) = a , then ω = P d − k =1 d − kd α k ∈ ∆. Given non-trivial φ ∈ ∆ and T ∈ SL ( d, R ),we define the φ -translation length ℓ φ ( T ) = φ ( ℓ ( T )) . Let (Σ + , σ ) be the Stadlbauer-Ledrappier-Sarig coding [35, 62] (if S has finite area) or Dal’bo-Peign´e coding [19] (if not) of the recurrent portion of the geodesic flow on T S . It is topologicallymixing and has (BIP). Moreover, it comes equipped with a map G : A →
Γso that if γ ∈ Γ is hyperbolic, then there exists x = x · · · x n ∈ Σ + so that G ( x ) · · · G ( x n ) isconjugate to γ . Moreover, x is unique up to powers of σ . Given a cusped Hitchin representation ρ : Γ → SL ( d, R ) we will define a vector-valued roof function τ ρ : Σ + → a with the property that if x = x · · · x n is a periodic element of Σ + , then S n τ ρ ( x ) = τ ρ ( x ) + τ ( σ ( x )) + · · · + τ ρ ( σ n − ( x )) = ℓ (cid:0) ρ ( G ( x ) · · · G ( x n )) (cid:1) so τ ρ encodes all the spectral data of ρ (Γ).The following result allows us to use the general thermodynamical machinery we developed tostudy cusped Hitchin representations. OUNTING AND EQUIDISTRIBUTION 5
Theorem D (Roof functions) . Suppose that Γ is a torsion-free, geometrically finite Fuchsian groupwhich is not convex cocompact, ρ : Γ → SL ( d, R ) is a cusped Hitchin representation and φ ∈ ∆ .Then there exists a locally H¨older continuous function τ φρ = φ ◦ τ ρ : Σ + → R such that (1) τ φρ is eventually positive and non-arithmetic. (2) If x = x · · · x n is a periodic element of Σ + , then S n τ φρ ( x ) = ℓ φ (cid:0) ρ ( G ( x ) · · · G ( x n )) (cid:1) . (3) τ φρ has a strong entropy gap at infinity. Moreover, if φ = a α + · · · + a d − α d − , then d ( τ φρ ) = 12( a + · · · + a d − ) . (4) If η : Γ → SL ( d, R ) is another cusped Hitchin representation, then there exists C > sothat | τ φρ ( x ) − τ φη ( x ) | ≤ C for all x ∈ Σ + . We obtain a counting result for cusped Hitchin representations as an immediate consequence ofTheorem A.
Corollary 1.3. If ρ : Γ → SL ( d, R ) is a cusped Hitchin representation and φ ∈ ∆ , then there existsa unique δ = δ φ ( ρ ) so that P ( − δτ φρ ) = 0 , and (cid:8) [ γ ] ∈ [Γ] (cid:12)(cid:12) < ℓ φ ( ρ ( γ )) ≤ t (cid:9) ∼ e tδ tδ as t → ∞ . We will refer to δ φ ( ρ ) as the φ -topological entropy of ρ .If ρ, η : Γ → SL ( d, R ) are cusped Hitchin representations and φ ∈ ∆, we define the Manhattancurve C φ ( ρ, η ) = { ( a, b ) ∈ R | P ( − aτ φρ − bτ φη ) = 0 , a ≥ , b ≥ , a + b > } . Theorem C immediately gives the following information about C φ ( ρ, η ). Corollary 1.4. If ρ, η : Γ → SL ( d, R ) are cusped Hitchin representations and φ ∈ ∆ , then (1) C φ ( ρ, η ) is a closed subsegment of an analytic curve, (2) the points ( δ φ ( ρ ) , and (0 , δ φ ( η )) lie on C φ ( ρ, η ) , (3) and C φ ( ρ, η ) is strictly convex, unless ℓ φ ( ρ ( γ )) = δ φ ( η ) δ φ ( ρ ) ℓ φ ( η ( γ )) for all γ ∈ Γ .Moreover, the tangent line to C φ ( ρ, η ) at ( h φ ( ρ ) , has slope s φ ( ρ, η ) = − R τ φη dµ − δ φ ( ρ ) τ φρ R τ φρ dµ − δ φ ( ρ ) τ φρ We call I φ ( ρ, η ) = − s φ ( ρ, η ) the φ -pressure intersection . We also define the renormalized φ -pressure intersection by J φ ( ρ, η ) = δ φ ( η ) δ φ ( ρ ) I φ ( ρ, η ) . As a further corollary of Theorem C we obtain the following rigidity result for renormalized pressureintersection. This corollary will later play a key role in our forthcoming construction of pressuremetrics on the space of cusped Hitchin representations.
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Corollary 1.5. If ρ, η : Γ → SL ( d, R ) are cusped Hitchin representations and φ ∈ ∆ , then J φ ( ρ, η ) ≥ with equality if and only if ℓ φ ( ρ ( γ )) = δ φ ( η ) δ φ ( ρ ) ℓ φ ( η ( γ )) for all γ ∈ Γ . As a corollary of Theorem B we obtain the following geometric interpretation of the pressureintersection. Let R φT ( ρ ) = (cid:8) [ γ ] ∈ [Γ] (cid:12)(cid:12) < ℓ φ ( ρ ( γ )) ≤ T (cid:9) . Corollary 1.6. If ρ, η : Γ → SL ( d, R ) are cusped Hitchin representations and φ ∈ ∆ then I φ ( ρ, η ) = lim T →∞ R φT ( ρ )) X [ γ ] ∈ R φT ( ρ ) ℓ φ ( η ( γ )) ℓ φ ( ρ ( γ )) . In a companion paper, Canary, Zhang and Zimmer [13] study the geometry of cusped Hitchinrepresentation showing that they are “relatively” Borel Anosov in a sense which generalizes workof Labourie [31]. They also show that cusped Hitchin representations are stable with respectto deformation in SL ( d, C ). As a consequence, they see that limit maps are H¨older and varyanalytically. In a sequel paper, we will combine the work in this paper and in [13] to constructpressure metrics on cusped Hitchin components.This project is motivated by the hope that there is an augmented Hitchin component whichgeneralizes the notion of augmented Teichm¨uller space. Masur [41] proved that the augmentedTeichm¨uller space is the metric completion of Teichm¨uller space with the Weil-Petersson metric.The strata at infinity of augmented Teichm¨uller space consists of Teichm¨uller spaces of cuspedhyperbolic surfaces. These strata naturally inherit a Weil-Petersson metric from the completion.The potential analogy is clearest when d = 3, where Hitchin components are spaces of convexprojective structures on closed surfaces. Work of Loftin [36] and Loftin-Zhang [37] explores theanalytic structure and topology of this bordification. We hope that our work on pressure metricswill aid in showing that there is an augmented Hitchin component which arises as the metriccompletion of the Hitchin component with the pressure metric. Other applications:
These results have immediate generalizations for P k -Anosov representationsof geometrically finite Fuchsian groups.We also recover (mild generalizations of) of many of Sambarino’s results on counting and equidis-tribution for uncusped Anosov representations in our framework (see [52, 53, 54]). Historical remarks:
Counting and equidistribution results have long been a central theme ofthe Thermodynamical Formalism (see, for example, the seminal work of Bowen, Parry, Pollicottand Ruelle [5, 6, 43, 50]). Lalley’s innovation [33] was the introduction of renewal theory and thedevelopment of a Renewal Theorem which allowed him to obtain precise counting and equidistribu-tion results. Our work harnesses Kesseb¨ohmer and Kombrink’s extension [30] of Lalley’s RenewalTheorem to the setting of countable Markov shifts to obtain similar results in our setting.Bishop and Steger [3] proved a rigidity theorem in the setting of finite area hyperbolic surfaceswhich is the precursor to the study of Manhattan curves. Lalley [34] extended Bishop and Steger’srigidity theorem to the setting of closed negatively curved surfaces. The formulation in terms of aManhattan curve is due to Burger [12] who worked in the setting of convex cocompact representa-tions into rank one Lie groups. Kao [25] established a Manhattan curve theorem for geometricallyfinite Fuchsian groups and Bray-Canary-Kao [9] extended his result to the setting of geometricallyfinite quasifuchsian representations.
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Dal’bo and Peign´e [19] used renewal theorems in their work obtaining counting and mixingresults on geometrically finite negatively curved surfaces. They also applied renewal techniques tostudy counting results for the modular surface [18]. Thirion [63] used related techniques to obtainasymptotic results for orbital counting functions for ping pong groups. Thirion’s ping pong groupsoverlap with the class of (images of) cusped P -Anosov representations.Corollary 1.3 generalizes results of Sambarino [52, 53, 54] from the Anosov setting, while Corol-laries 1.5 and 1.6 generalize results of Bridgeman-Canary-Labourie-Sambarino [10].In the case of cusped Hitchin representations, d ( τ φρ ) is simply the maximum critical exponentof the φ -length Poincar´e series associated to any unipotent subgroup of ρ (Γ). Thus, having astrong entropy gap at infinity is analogous to the critical exponent gap used in the work of Dal’bo-Peign´e [19] and Dal’bo-Otal-Peign´e [17]. Schapira and Tapie [60, Prop. 7.16] showed that for ageometrically finite negatively curved manifold then there is a critical exponent gap if and onlyif the geodesic flow has an entropy gap at infinity. Our definition is inspired by their work. Inturn, Schapira and Tapie were motivated, in part, by work on strongly positive recurrent potentialsfor countable Markov shifts due to Gurevich-Savchenko [23], Sarig [56, 57], Ruette [51], and Boyle-Buzzi-G´omez [8]. Other relevant precursors to our results include the work Iommi-Riquelme-Velozo[24], Riquelme-Velozo [49], and Velozo [65].In recent work, Pollicott and Urbanski [46] use related techniques to obtain fine counting resultsfor conformal dynamical systems. Their main technical tools come from the study of complexifiedRuelle-Perron-Frobenius operators, generalizing early work of Parry-Pollicott [43] in the settingof finite Markov shifts. (The proof of Kesseb¨ohmer and Kombrink’s Renewal Theorem [30] alsorelies on the study of complexified Ruelle-Perron-Frobenius operators.) Pollicott and Urbanskigive extensive applications to the study of circle packings, rational functions, continued fractions,Fuchsian groups and Schottky groups and other topics.Feng Zhu [67] obtained closely related counting and equidistribution results for the Hilbert lengthfunctional on geometrically finite strictly convex projective manifolds. When d = 3, cusped Hitchinrepresentations are holonomy maps of strictly convex projective surfaces, so our results overlapwith his in this case. Outline of paper:
In Section 2, we recall the relevant background material from the theory ofcountable Markov shifts. In Section 3, we use this theory to explore the consequences of entropygaps at infinity. In Section 4, we recall the Renewal Theorem of Kesseb¨ohmer and Kombrink [30]and show that we can apply it in our context. Section 5 contains the crucial technical materialneeded in the proof of Theorems A . Sections 6, 7 and 8 contain the proof of Theorems A, Band C (respectively). In Section 9, we develop the background material needed for our applica-tions. Section 10 contains the proof of (a generalization of) Theorem D and section 11 derives itsconsequences.
Acknowledgements:
The authors would like to thank Godofredo Iommi, Andres Sambarino,Barbara Schapira, Ralf Spatzier and Dan Thompson for helpful comments and suggestions.This material is based upon work supported by the National Science Foundation under GrantNo. DMS-1928930 while the second author participated in a program hosted by the MathematicalSciences Research Institute in Berkeley, California, during the Fall 2020 semester.2.
Background from the Thermodynamic Formalism
In this section, we recall the background results we will need from the Thermodynamic Formalismfor countable Markov shifts as developed by Gurevich-Savchenko [23], Mauldin-Urbanksi [42] andSarig [56].Given a countable alphabet A and a transition matrix T = ( t ab ) ∈ { , } A×A a one-sided Markovshift is Σ + = { x = ( x i ) ∈ A N | t x i x i +1 = 1 for all i ∈ N } OUNTING AND EQUIDISTRIBUTION 8 equipped with a shift map σ : Σ + → Σ + which takes ( x i ) i ∈ N to ( x i +1 ) i ∈ N .We will work in the setting of topologically mixing Markov shifts with (BIP), where many of theclassical results of Thermodynamic formalism generalize. The shift (Σ + , σ ) is topologically mixing if for all a, b ∈ A , there exists N = N ( a, b ) so that if n ≥ N , then there exists x ∈ Σ so that x = a and x n = b . It has the big images and pre-images property (BIP) if there exists a finite subset B ⊂ A so that if a ∈ A , then there exist b , b ∈ B so that t b a = 1 = t ab .The theory works best for locally H¨older continuous potentials. We say that g : Σ + → R is locally H¨older continuous if there exist A > α > | g ( x ) − g ( y ) | ≤ Ae − αn whenever x i = y i for all i ≤ n . When we want to record the constants we will say that g is locally α -H¨older continuous with constant A . The Gurevich pressure of g is given by P ( g ) = lim n →∞ n log X { x ∈ Fix n | x = a } e S n g ( x ) for some (any) a ∈ A where S n g ( x ) = n X i =1 g ( σ i − ( x ))is the ergodic sum and Fix n = { x ∈ Σ + | σ n ( x ) = x } .We say that two locally H¨older continuous functions f and g are cohomologous if there exists alocally H¨older continuous function h so that f − g = h − h ◦ σ. The analogue of Livsic’s theorem holds in this setting.
Theorem 2.1. (Sarig [59, Thm 1.1])
Suppose that Σ + is a topologically mixing, one-sided countableMarkov shift with (BIP). If f : Σ + → R and g : Σ + → R are both locally H¨older continuous, then f is cohomologous to g if and only if S n f ( x ) = S n g ( x ) for all n ∈ N and x ∈ Fix n . In particular,if f and g are cohomologous, then P ( − tf ) = P ( − tg ) whenever P ( − tf ) is finite. A σ -invariant Borel probability measure µ on Σ + is an equilibrium state for a locally H¨oldercontinuous function g : Σ + → R if P ( g ) = h σ ( µ ) + Z Σ + g dµ where h σ ( µ ) is the measure-theoretic entropy of σ with respect to the measure µ .A Borel probability measure µ on Σ + is a Gibbs state for a locally H¨older continuous function g : Σ + → R if there exists B > B ≤ µ ([ a , . . . , a n ]) e S n g ( x ) − nP ( g ) ≤ B for all x ∈ [ a , . . . , a n ], where [ a , . . . , a n ] is the cylinder consisting of all x ∈ Σ + so that x i = a i forall 1 ≤ i ≤ n . Theorem 2.2. (Mauldin-Urbanski [42, Thm 2.2.9], Sarig [59, Thm 4.9]) If Σ + is a topologicallymixing, one-sided countable Markov shift with (BIP), g : Σ + → R is locally H¨older continuous, itadmits a shift invariant Gibbs state µ g , and − R g dµ g < + ∞ , then µ g is the unique equilibriumstate for g . Recall from the introduction that for g : Σ + → R a locally H¨older continuous function we define I ( g, a ) = inf (cid:8) g ( x ) | x ∈ Σ + , x = a (cid:9) and S ( g, a ) = sup (cid:8) g ( x ) | x ∈ Σ + , x = a (cid:9) . We will make crucial use of the following criterion for a potential to admit an equilibrium state.
OUNTING AND EQUIDISTRIBUTION 9
Theorem 2.3. (Mauldin-Urbanski [42, Thm 2.2.4 and 2.2.9, Lemma 2.2.8], Sarig [59, Thm 4.9]) If Σ + is a topologically mixing, one-sided countable Markov shift with (BIP), g : Σ + → R is locallyH¨older continuous, and X a ∈A I ( g, a ) e − S ( g,a ) converges, then − g admits an unique equilibrium state µ − g . Moreover, Z Σ + g dµ − g < + ∞ . We will need to be able to take the derivatives of the pressure function and to be able to applythe Implicit Function Theorem. We say that { g u : Σ + → R } u ∈ M is a real analytic family if M isa real analytic manifold and for all x ∈ Σ + , u → g u ( x ) is a real analytic function on M . Mauldinand Urbanski [42, Thm. 2.6.12, Prop. 2.6.13] (see also Sarig [58, Cor. 4]), prove real analyticityproperties of the pressure function and evaluate its derivative. Theorem 2.4. (Mauldin-Urbanski, Sarig)
Suppose that Σ + is a topologically mixing, one-sidedcountable Markov shift with (BIP). If { g u : Σ + → R } u ∈ M is a real analytic family of locally H¨oldercontinuous functions such that P ( g u ) < ∞ for all u , then u → P ( g u ) is real analytic.Moreover, if v ∈ T u M and there exists a neighborhood U of u in M so that if u ∈ U and − R Σ + g u dµ g u < ∞ , then D v P ( g u ) = Z Σ + D v ( g u ( x )) dµ g u . Recall that if f : Σ + → R is locally H¨older continuous the transfer operator is defined by L f φ ( x ) := X y ∈ σ − ( x ) e f ( y ) φ ( y )where φ : Σ + → R is a bounded locally H¨older continuous function. The transfer operator, inparticular, gives us crucial information about equilibrium states. Theorem 2.5. (Mauldin-Urbanski [42, Cor. 2.7.5], Sarig [59, Thm. 4.9])
Suppose that Σ + is atopologically mixing, one-sided countable Markov shift with (BIP). If g : Σ + → R is locally H¨oldercontinuous, P ( g ) < + ∞ , and sup g < + ∞ then there exist unique probability measures µ g and ν g on Σ + and a positive function h g : Σ + → R so that µ g = h g ν g , L g h g = e P ( g ) h g , and L ∗ g ν g = e P ( g ) ν g . Moreover, h g is bounded away from both and + ∞ and µ g is an equilibrium state for g . We will also use the following estimate on the behavior of powers of the transfer operator.
Theorem 2.6. (Mauldin-Urbanski [42, Theorem 2.4.6])
Suppose that Σ + is a topologically mix-ing, one-sided countable Markov shift with (BIP). If g : Σ + → R is locally H¨older continuous, P ( g ) < + ∞ , and sup g < + ∞ , then there exist R > and η ∈ (0 , so that if n ∈ N and φ : Σ + → R is bounded and locally η -H¨older continuous with constant A , then (2.1) (cid:13)(cid:13)(cid:13) e − nP ( g ) L ng φ − h g ( x ) Z φ dν g (cid:13)(cid:13)(cid:13) ≤ Rη n (cid:16) sup x ∈ Σ + | φ ( x ) | + A (cid:17) . Entropy gaps at infinity
In this section, we show that a strong entropy gap at infinity implies a weak entropy gap atinfinity and explore the thermodynamical consequences of entropy gaps at infinity.
OUNTING AND EQUIDISTRIBUTION 10
Recall that d ( f ) is the critical exponent of the series Z ( f, s ) = X a ∈A e − sS ( f,a ) . Notice that if f is locally H¨older continuous, there exists C > S ( f, a ) − I ( f, a ) ≤ C forall a ∈ A . So the series X a ∈A e − sI ( f,a ) has critical exponent d ( f ) and diverges at d ( f ) if and only if f has a strong entropy gap at infinity.We first observe a bound on the number of letters with I ( f, a ) ≤ t . Lemma 3.1.
Suppose that Σ + is a topologically mixing, one-sided countable Markov shift with(BIP). If f : Σ + → R is locally H¨older continuous, d ( f ) is finite and b > d ( f ) , then there exists D = D ( f, b ) > so that B ( f, t ) = (cid:8) a ∈ A | I ( f, a ) ≤ t (cid:9) ≤ De bt for all t > , and X y ∈ σ − ( x ) { f ( y ) ≤ t } ( y ) ≤ De bt for all x ∈ Σ + and t > .Proof. Fix b > d ( f ). If there does not exist D so that B ( f, t ) ≤ De bt for all t >
0, then thereexists a sequence t n → ∞ so that B ( f, t n ) ≥ ne bt n . But then X a ∈A e − bI ( f,a ) ≥ X { a | I ( f,a ) ≤ t n } e − bI ( f,a ) ≥ ne bt n e − bt n = n which contradicts our assumption that b > d ( f ).Finally, notice that if x ∈ Σ + , then X y ∈ σ − ( x ) { f ( y ) ≤ t } ( y ) ≤ B ( f, t ) ≤ De bt for all t > (cid:3) It will often be convenient to work with a strictly positive potential. We observe that an eventu-ally positive potential is always cohomologous to a strictly positive potential with the same entropygaps.
Lemma 3.2.
Suppose that Σ + is a topologically mixing, one-sided countable Markov shift with(BIP) and that f : Σ + → R is eventually positive, locally H¨older continuous and d ( f ) is finite.Then f is cohomologous to a strictly positive, locally H¨older continuous function g so that (1) there exists C so that | f ( x ) − g ( x ) | ≤ C for all x ∈ Σ + , (2) d ( f ) = d ( g ) , (3) f has a weak entropy gap at infinity if and only if g has a weak entropy gap at infinity, and (4) f has a strong entropy gap at infinity if and only if g has a strong entropy gap at infinity.Proof. Notice that (1) implies that | I ( f, a ) − I ( g, a ) | ≤ C . Moreover, if f is cohomologous to g ,and both are locally H¨older continuous, then P ( − tf ) = P ( − tg ) for all t > d ( f ), see Theorem2.1. Therefore, (2)–(4) follow immediately once we construct a strictly positive, locally H¨oldercontinuous function g that is cohomologous to f so that (1) holds.Let R = (cid:12)(cid:12)(cid:12)(cid:12) inf x ∈ Σ + f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . OUNTING AND EQUIDISTRIBUTION 11
Note that R = | inf a ∈A I ( f, a ) | is finite since there exists s > d ( f ) > P a ∈A e − sI ( f,a ) isfinite. Since f is eventually positive, there exists N ∈ N and B > n ≥ N and x ∈ Σ + ,then S n f ( x ) ≥ B. Let F = { a ∈ A | I ( f, a ) ≤ RN + B } . Since d ( f ) is finite, F must be finite. To see this, observe that for s > d ( f ) > ∞ > X a ∈A e − sI ( f,a ) ≥ X a ∈F e − sI ( f,a ) ≥ X a ∈F e − s ( RN + B ) . For all n ∈ N , define C n f ( x ) = n X i =1 (cid:16) f ( σ i − ( x )) { x i ∈F} ( x ) + ( RN + B ) { x i / ∈F} ( x ) (cid:17) = S n f ( x ) − n X i =1 (cid:16) f ( σ i − ( x )) − ( RN + B ) (cid:17) { x i ( x ) . By construction, RN + N B + T N ≥ C N f ( x ) ≥ B for all x ∈ Σ + , where T = sup { f ( x ) | x ∈ F } . (The lower bound holds, since C N f ( x ) = S N f ( x ) ≥ B if x i ∈ F for all i ≤ N , and otherwise one ofthe summands of C N f ( x ) is RN + B and each of the remaining terms are bounded below by − R .)We then define g : Σ + → R by g ( x ) = 1 N C N f ( x ) + (cid:0) f ( x ) − ( RN + B ) (cid:1) { x ( x ) . By construction, g is continuous and g ( x ) ≥ BN > x ∈ Σ + , so g is strictly positive.Moreover, if x ∈ F , then | g ( x ) − f ( x ) | ≤ RN + B + 2 T , and if x / ∈ F , then | g ( x ) − f ( x ) | ≤ RN + B + N C N f ( x ) ≤ RN + B ). It follows that | g ( x ) − f ( x ) | ≤ RN + B + T ) =: C for all x ∈ Σ + .To show g is locally H¨older continuous, consider x, y ∈ Σ + for which x i = y i for all i = 1 , . . . , n ,and note that it suffices to consider n ≥ N . Then | g ( x ) − g ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 ( f ( σ i − ( x )) − f ( σ i − ( y ))) { x i ∈F} ( x ) ! + ( f ( x ) − f ( y )) { x ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since n ≥ N, applying local H¨older continuity of f gives the desired conclusion.Finally, if x = x . . . x r ∈ Fix r , then one may check that S r f ( x ) = S r g ( x ). To see this, observethat S r g ( x ) = S r (cid:18) N C N f ( x ) (cid:19) + S r (cid:16) ( f ( x ) − ( RN + B )) { x ( x ) (cid:17) = 1 N S r C N f ( x ) + r X j =1 (cid:0) f ( σ j − ( x )) − ( RN + B ) (cid:1) { x j ( x ) OUNTING AND EQUIDISTRIBUTION 12 and since σ r ( x ) = x , S r C N f ( x ) = S r S N f ( x ) − r X j =1 N X i =1 (cid:0) f ( σ i − ( x )) − ( RN + B ) (cid:1) { x i ( x )= N S r f ( x ) − N r X j =1 (cid:0) f ( σ j − ( x )) − ( RN + B ) (cid:1) { x j ( x ) . Theorem 2.1 then implies that f and g are cohomologous. (cid:3) We next study the behavior of P ( − tf ) for t > d ( f ), showing among other things that a strongentropy gap at infinity implies a weak entropy gap at infinity. Lemma 3.3.
Suppose that Σ + is a topologically mixing, one-sided countable Markov shift with(BIP) and f : Σ + → R is locally H¨older continuous and eventually positive. (1) If d ( f ) is finite, then P ( − tf ) is finite if t > d ( f ) and infinite if t < d ( f ) , and the function t → P ( − tf ) is monotone decreasing and analytic on ( d ( f ) , ∞ ) . (2) There exists at most one δ ∈ ( d ( f ) , ∞ ) so that P ( − δf ) = 0 . (3) If f has a strong entropy gap at infinity, then t → P ( − tf ) is proper on ( d ( f ) , ∞ ) . Inparticular, f has a weak entropy gap at infinity.Proof. Mauldin and Urbanski [42, Theorem 2.1.9] proved that if Σ + is topologically mixing andhas (BIP), then P ( − sf ) is finite if and only if Z ( − f, s ) = X a ∈A e sup {− sf ( x ) | x = a } converges. Therefore, P ( − tf ) is finite if t > d ( f ) and infinite if t < d ( f ). Notice that t → P ( − tf ) ismonotone decreasing by definition and analytic by Theorem 2.4, so (1) follows. (2) is an immediateconsequence of (1).It remains to show (3). The fact that lim t → d ( f ) P ( − tf ) = + ∞ is essentially contained in Mauldinand Urbanski’s proof of [42, Theorem 2.1.9], but we elaborate here for completeness. They showthat there exist constants q, s, M, m > g , n + s ( n − X i = n Z i ( g, ≥ e − M +( M − m ) n q n − Z ( g, n . where Z n ( g,
1) = X p ∈ Λ k e sup x ∈ p S n g ( x ) , and Λ k is the set of k -cylinders of Σ + . They observe [42, Equation (2.1)] that lim n log Z n ( g,
1) = P ( g ).Thus there exists A > n , there exists ˆ n ∈ [ n, n + s ( n − Z ˆ n ( g, ≥ A n Z ( g, n , so P ( g ) ≥ s log AZ ( g, f has a strong entropy gap at infinity, thenlim t → d ( f ) Z ( − tf,
1) = + ∞ and hencelim t → d ( f ) P ( − tf ) ≥ lim t → d ( f )
11 + s log AZ ( − tf,
1) = + ∞ . We now show that lim t →∞ P ( − tf ) = −∞ . Notice that since there exists N > S n f ( x ) > B > n ≥ N and x ∈ Σ + , we have S kN f ( x ) > kB for every k ≥
1. Then, X { x ∈ Fix kN | x = a } e − td ( f ) S kN f ( x ) ≤ X { x ∈ Fix kN | x = a } e − t − d ( f ) kB − d ( f ) S kN f ( x )OUNTING AND EQUIDISTRIBUTION 13 which implies P ( − td ( f ) f ) ≤ lim k →∞ kN log X { x ∈ Fix kN | x = a } e − t − d ( f ) kB − d ( f ) S kN f ( x ) = − t − d ( f ) BN + P ( − d ( f ) f )and so lim t →∞ P ( − tf ) = −∞ .Since t → P ( − tf ) is proper and monotone decreasing on ( d ( f ) , ∞ ), it follows that there exists δ > d ( f ) so that P ( − δf ) = 0. Therefore, f has a weak entropy gap at infinity and we haveestablished (3). (cid:3) We next notice that − tf admits an equilibrium state if t > d ( f ). Lemma 3.4.
Suppose that Σ + is a topologically mixing, one-sided countable Markov shift with(BIP). If f : Σ + → R is locally H¨older continuous and eventually positive and t > d ( f ) , then thereexists a unique equilibrium state µ − tf for − tf . Moreover, < Z Σ + f dµ − tf < + ∞ . Proof.
Theorem 2.3 implies that there exists a unique equilibrium state for − tf if and only if X a ∈A tI ( f, a ) e − tS ( f,a ) < + ∞ . Indeed, this series converges since X a ∈A e − sS ( f,a ) < + ∞ for all s > d ( f ). Theorem 2.3 also ensures that R Σ + f dµ − tf < + ∞ . Since f is eventually positive,it is cohomologous to a strictly positive function g . Then − tf and − tg are cohomologous andhence have the same integral with respect to any shift-invariant measure, and also share the sameshift-invariant equilibrium state, i.e. µ − tf = µ − tg (see [42, Theorem 2.2.7] and Theorem 2.3).Hence, Z Σ + f dµ − tf = Z Σ + g dµ − tg > . (cid:3) Theorem 2.5 and Lemma 3.3 have the following corollary which we will use repeatedly.
Corollary 3.5.
Suppose that Σ + is a topologically mixing, one-sided countable Markov shift with(BIP). If f : Σ + → R is locally H¨older continuous, eventually positive, and has a weak entropy gapat infinity and t > d ( f ) , then there exist unique probability measures µ − tf and ν − tf on Σ + and apositive function h − tf : Σ + → R so that µ − tf = h − tf ν − tf , L − tf h − tf = e P ( − tf ) h − tf , and L ∗− tf ν − tf = e P ( − tf ) ν − tf and h − tf is bounded away from both and + ∞ . Moreover, µ − tf is the equilibrium state of − tf . We will need analogues of these results for functions of the form − zg − δf where g is comparableto f and z is close to 0. Proposition 3.6.
Suppose that Σ + is a topologically mixing, one-sided countable Markov shift with(BIP), f : Σ + → R is locally H¨older continuous, eventually positive and has a weak entropy gapat infinity and P ( − δf ) = 0 for δ = δ ( f ) > d ( f ) > . If g : Σ + → R is locally H¨older continuous,eventually positive, and there exists C so that | f ( x ) − g ( x ) | ≤ C for all x ∈ Σ + , then OUNTING AND EQUIDISTRIBUTION 14 (1) if z > d ( f ) − δ , then P ( − zg − δf ) is finite, z → P ( − zg − δf ) is monotone decreasing andanalytic on ( d ( f ) − δ, ∞ ) and sup x ∈ Σ + ( − zg − δf ) < + ∞ . (2) if z > d ( f ) − δ , then there exist unique probability measures µ − zg − δf and ν − zg − δf on Σ + and a positive function h − zg − δf : Σ + → R so that µ − zg − δf = h − zg − δg ν − zg − δf , L − zg − δf h − zg − δf = e P ( − zg − δf ) h − zg − δff , and L ∗− zg − δf ν − zg − δf = e P ( − zg − δf ) ν − zg − δf . Moreover, h − zg − δf is bounded away from both and + ∞ and µ − zg − δf is the unique equi-librium state of − zg − δf .Proof. Notice that X { x ∈ Fix n | x = a } e S n ( − zg − δf ) ≤ X { x ∈ Fix n | x = a } e nzC e S n (cid:0) ( − z − δ ) f (cid:1) so P ( − zg − δf ) is finite if z + δ > d ( f ), i.e. if z > d ( f ) − δ . Similarly, if x ∈ Σ + , then( − zg − δf )( x ) ≤ − ( z + δ ) f ( x ) + Cz ≤ sup( − ( z + δ ) f ) + Cz < + ∞ if z + δ >
0. The function z → P ( − zg − δf ) is monotone decreasing by definition and analytic byTheorem 2.4. We have established (1).(2) is then an immediate consequence of (1) and Theorem 2.5. (cid:3) Renewal Theorems
Our main tool will be the Renewal Theorem of Kesseb¨ohmer and Kombrink [30]. Their resultgeneralized a result of Lalley [33] for finite Markov shifts.Consider a locally H¨older continuous potential f : Σ + → R . If φ : Σ + → R is a non-negative,bounded, locally H¨older continuous function, we define the renewal function N f ( φ, x, t ) := ∞ X n =0 X y ∈ σ − n ( x ) φ ( y ) { S n f ( y ) ≤ t } ( y ) . We recall that N f ( φ, x, t ) satisfies the renewal equation (4.1) N f ( φ, x, t ) = X y ∈ σ − ( x ) N f ( φ, y, t − f ( y )) + φ ( x ) { t ≥ } ( t ) Theorem 4.1. (Renewal theorem; Kesseb¨ohmer-Kombrink [30, Theorem 3.1])
Suppose that Σ + isa topologically mixing, one-sided, countable Markov shift with (BIP) and f : Σ + → R is a strictlypositive, non-arithmetic, locally H¨older continuous function so that there exists a unique δ > sothat P ( − δf ) = 0 and an equilibrium state µ − δf = h − δf ν − δf for − δf so that R Σ + tf dµ − δf < + ∞ for all t in some neighborhood of δ .If φ : Σ + → R is non-negative, bounded, not identically zero, and locally H¨older continuous andthere exists c > such that N f ( φ, x, t ) ≤ ce tδ , then N f ( φ, x, t ) ∼ e tδ δ h − δf ( x ) R Σ + φ dν − δf R Σ + f dµ − δf as t → ∞ , uniformly for x ∈ Σ + . OUNTING AND EQUIDISTRIBUTION 15
Remark . The Renewal Theorem we state above is a special case of [30, Theorem 3.1 (i)].Following the notations in [30], in our case η = 0 and f y ( t ) = ( t ≥
00 otherwise . Kesseb¨ohmer andKombrink [30] in place of our assumption of non-arithmeticity only require the weaker assumptionthat f is not a lattice, i.e. that f is not cohomologous to a function so that { S n f ( x ) | x ∈ Σ + } does not lie in a discrete subgroup of R . Moreover, since f y ( t ) ≥ R ∞−∞ e − T δ f y ( T ) dT = δ , and N f ( φ, x, t ) = 0 for t < f is strictly positive, their conditions (B) and (D) are satisfied. So,it only remains to check that their condition (C) is satisfied, which translates to the existence of c > N f ( φ, x, t ) ≤ ce tδ . We first check that a weak entropy gap at infinity implies such a bound on N f ( , x, t ). Lemma 4.3.
Suppose that Σ + is a topologically mixing, one-sided, countable Markov shift with(BIP) and f : Σ + → R is a strictly positive, locally H¨older continuous function with a weak entropygap at infinity. Let δ > d ( f ) be the unique constant such that P ( − δf ) = 0 . Then there exists C > such that N f ( , x, t ) = ∞ X n =0 X y ∈ σ − n ( x ) { S n f ( y ) ≤ t } ( y ) ≤ Ce tδ for all x ∈ Σ + and t > . We adopt the strategy of Lalley [33, Lemma 8.1].
Proof.
Define for all x ∈ Σ + and t > G ( x, t ) = e − tδ N f ( , x, t ) h − δf ( x )where h − δf is the eigenfunction for the transfer operator given by Theorem 2.5. Let b G ( t ) = sup { G ( x, s ) | x ∈ Σ + , s ≤ t } . Notice that b G ( t ) is finite for all t >
0, since h − δf is bounded away from 0, and for any fixed t > a ∈ A so that I ( f, a ) ≤ t (which implies that there are only finitelymany n and only finitely many y ∈ σ − n ( x ), for each n , so that S n f ( y ) ≤ t ). Since h − δf is boundedaway from 0 and ∞ , it remains to show that there exists ˆ C so that b G ( t ) ≤ ˆ C for all t > G ( x, t ) = X y : σ ( y )= x G ( y, t − f ( y )) e − δf ( y ) h − δf ( y ) h − δf ( x ) + e − tδ h − δf ( x ) . for all t >
0. Notice that since h − δf ( x ) is the eigenfunction of L − δf with eigenvalue 1 = e P ( − δf ) , X y : σ ( y )= x e − δf ( y ) h − δf ( y ) h − δf ( x ) = ( L − δf h − δf ) ( x ) h − δf ( x ) = 1 . If c = c ( f ) = inf x ∈ Σ + f ( x ) >
0, then(4.2) G ( x, t ) ≤ b G ( t − c ) + e − tδ h − δf ( x ) OUNTING AND EQUIDISTRIBUTION 16 for all x ∈ Σ + and t ≥ c . Therefore, b G ( mc ) ≤ b G ( c ) + ˆ H m X n =1 e − cnδ for all m ∈ N , where ˆ H = sup n h − δf ( x ) | x ∈ Σ + o . Since b G is increasing, b G ( t ) ≤ ˆ C = b G ( c ) + ˆ H ∞ X n =1 e − cnδ for all t >
0, which completes the proof. (cid:3) If φ : Σ + → R is bounded, non-negative and locally H¨older continuous, then N f ( φ, x, t ) ≤ (cid:16) sup x ∈ Σ + φ ( x ) (cid:17) N f ( , x, t ) , so Lemmas 3.3, 3.4 and 4.3 together imply that we can apply the Renewal Theorem to φ when f is strictly positive and has a weak entropy gap at infinity. Corollary 4.4.
Suppose that Σ + is a topologically mixing, one-sided, countable Markov shift with(BIP) and f : Σ + → R is a strictly positive, non-arithmetic, locally H¨older continuous functionwith a weak entropy gap at infinity, P ( − δf ) = 0 and h − δf ν − δf = µ − δf is the equilibrium state for − δf . If φ : Σ + → R is bounded, non-negative, not identically zero and locally H¨older continuous,then N f ( φ, x, t ) ∼ e tδ δ h − δf ( x ) R Σ + φ dν − δf R Σ + f dµ − δf as t → ∞ , uniformly for x ∈ Σ + . Preparing to count
In this section we develop the technical tools needed in the proofs of our counting result. Themajority of these results bound the size of various subsets of the shift space. Most importantly, weshow that if y ∈ σ n ( x ) and S n f ( y ) is “large,” then “typically” S n f ( y ) is close to n R Σ + f dµ − δf .These results and their proofs generalize Lalley [33, Theorem 6]. The fact that our Markov shift iscountable requires more delicate control of error estimates.For each cylinder p , we choose a sample point z p ∈ p which is not periodic. We then define W ( n, p, t ) = X y ∈ σ − n ( z p ) p ( y ) { x | S n f ( x ) ≤ t } ( y ) = (cid:16) p ∩ σ − n ( z p ) ∩ { x | S n f ( x ) ≤ t } (cid:17) . We show that the W ( n, p, t ) may be used to approximate the size of M f ( n, t ). This allows us toreplace the counting of fixed points with counting of pre-images of our sample points.If k ∈ N , let Λ k be the countable partition of Σ + into k -cylinders. Lemma 5.1.
Suppose that Σ + is a topologically mixing, one-sided countable Markov shift with(BIP), f : Σ + → R is locally H¨older continuous strictly positive and has a weak entropy gap atinfinity. If P ( − δf ) = 0 and µ − δf is the equilibrium state for − δf , then (i) If v k = inf { µ − δf ( p ) | p ∈ Λ k } , then lim k →∞ v k = 0 . (ii) For any p ∈ Λ k and n ≥ k there exists a bijection Ψ np : Fix n ∩ p → σ − n ( z p ) ∩ p. OUNTING AND EQUIDISTRIBUTION 17 (iii)
There exists a sequence { ǫ k } such that lim ǫ k = 0 and if y ∈ Fix n ∩ p and n ≥ k , then | S n f ( y ) − S n f (Ψ np ( y )) | ≤ ǫ k . (iv) If n ≥ k , then (5.1) X p ∈ Λ k W ( n, p, t − ǫ k ) ≤ M f ( n, t ) ≤ X p ∈ Λ k W ( n, p, t + ǫ k ) . Moreover, for all k ∈ N and s ∈ ( d ( f ) , δ ) , there exists C ( k, s ) > such that for any n < k and t > , X p ∈ Λ k W ( n, p, t ) ≤ C ( k, s ) e st and M f ( k, t ) ≤ C ( k, s ) e st . Proof.
Recall that since µ − δf is a Gibbs state for − δf (see Theorem 2.2) and P ( − δf ) = 0, thereexists B > p ∈ Λ k , and x ∈ pµ − δf ( p ) ≤ Be − δS k f ( x ) . Since f is strictly positive, lim k →∞ inf { S k ( x ) | x ∈ Σ + } = + ∞ , so (i) holds.Given p ∈ Λ k , we define an explicit bijectionΨ np : Fix n ∩ p → σ − n ( z p ) ∩ p If y = y y ...y n ∈ Fix n ∩ p , then letΨ np ( y ) = y · · · y n z · · · z m · · · . Notice that since y = z and y · · · y n ∈ Σ + , we must have t y n y = t y n z = 1, so Ψ np ( y ) ∈ Σ + . Themap Ψ np is injective by definition. If x ∈ σ − n ( z p ) ∩ p , then, since n ≥ k , x n +1 = z = x , whichimplies that x · · · x n ∈ Fix n ∩ p , so Ψ np is also surjective. Thus, we have established (ii).Since f is locally H¨older continuous, there exists B > r ∈ (0 ,
1) so that | f ( x ) − f ( y ) | ≤ Br l if x i = y i for all i ≤ l . Therefore, if y ∈ Fix n ∩ p , then, since z p ∈ p , y i = Ψ np ( y ) i for all i ≤ n + k , so | S n f ( y ) − S n f (Ψ np ( y )) | ≤ ǫ k = B ∞ X l = k r l . The first statement in (iv) follows immediately from (ii) and (iii). Choose b ∈ ( d ( f ) , z ). Lemma3.1 implies that there exists D so that B ( f, t ) = (cid:8) a ∈ A | I ( f, a ) ≤ t (cid:9) ≤ De bt . If c = c ( f ) = inf x ∈ Σ + f ( x ) = inf a ∈A I ( f, a ) > r ∈ N , then B ( f, rc ) = (cid:8) ( a , a ) ∈ A × A | I ( f, a ) + I ( f, a ) ≤ rc (cid:9) ≤ r X s =1 B ( f, rc − sc ) B ( f, sc ) ≤ r X s =1 D e brc = rD e brc . We may use the argument above to inductively show that B k ( f, rc ) = n ( a i ) ∈ A k (cid:12)(cid:12)(cid:12) k X i =1 I ( f, a i ) ≤ rc o ≤ r k − D k e brc . OUNTING AND EQUIDISTRIBUTION 18
Notice that X p ∈ Λ k W ( n, p, rc ) ≤ B n ( f, rc ) and M f ( k, rc ) ≤ B k ( f, rc )so (iv) follows. (cid:3) We set up some convenient notation. If x ∈ Σ + , let W ( x, t ) = n y ∈ Σ + (cid:12)(cid:12)(cid:12) σ n ( y ) = x, S n f ( y ) ≤ t for some n ≥ o Observe that if x is not periodic and y ∈ W ( x, t ), then there is a unique n ( y ) so that σ n ( y ) ( y ) = x .If x is not periodic and ǫ >
0, we let W ( x, t, ≤ ǫ ) = n y ∈ W ( x, t ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) tn ( y ) − ¯ f (cid:12)(cid:12)(cid:12) ≤ ǫ o , and W ( x, t, > ǫ ) = n y ∈ W ( x, t ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) tn ( y ) − ¯ f (cid:12)(cid:12)(cid:12) > ǫ o = W ( x, t ) − W ( x, t, ≤ ǫ )where ¯ f = R Σ + f dµ − δf . Moreover, let W ( x, t ) = W ( x, t ) , W ( x, t, < ǫ ) = W ( x, t, ≤ ǫ ) and W ( x, t, > ǫ ) = W ( x, t, > ǫ ) = W ( x, t ) − W ( x, t, ≤ ǫ ) . The crucial technical result we need for the proof of our counting result is a uniform bound onthe growth of W ( x, t, > ǫ ). Proposition 5.2.
Suppose that Σ + is a topologically mixing, one-sided, countable Markov shiftwith (BIP) and f : Σ + → R is a strictly positive, locally H¨older continuous function with a weakentropy gap at infinity. Let δ > d ( f ) be the unique constant such that P ( − δf ) = 0 . Given ǫ > ,there exist D > and b < δ so that W ( x, t, > ǫ ) ≤ De bt for any non-periodic x ∈ Σ + .Proof. Fix, for the entire proof, ǫ ∈ (0 , ¯ f / s > d ( f ), then there exist R s > η s ∈ (0 ,
1) so that(5.2) (cid:13)(cid:13)(cid:13) e − nP ( − sf ) L n − sf ( x ) − h − sf ( x ) Z dν − zf (cid:13)(cid:13)(cid:13) ≤ R s η ns . If s > δ , then P ( − sf ) <
0, since P ( − δf ) = 0 and s → P ( − sf ) is monotone decreasing andcontinuous on ( d ( f ) , ∞ ) (by Lemma 3.3). Then, for any m ∈ N and t > X n ≥ m X y ∈ σ − n ( x ) { S n f ( y ) ≤ t } ( y ) ≤ X n ≥ m X y ∈ σ − n ( x ) e − s ( S n f ( y ) − t ) = e st X n ≥ m (cid:0) L n − sf (cid:1) ( x ) ≤ e st X n ≥ m e nP ( − sf ) ( h − sf ( x ) + R s η ns ) ≤ e st (cid:16) e mP ( − sf ) − e P ( − sf ) (cid:0) H s + R s (cid:1)(cid:17) . where H s = sup { h − sf ( x ) | x ∈ Σ + } . OUNTING AND EQUIDISTRIBUTION 19 If tn ( y ) − ¯ f < − ǫ , then n ( y ) ¯ f > t + n ( y ) ǫ and n ( y ) > t ¯ f − ǫ , so n ( y ) ¯ f > t (1 + ǫ ) where ǫ = ǫ ¯ f − ǫ .Given t >
0, let m t = j t (1+ ǫ )¯ f k . Then (cid:8) y ∈ W ( x, t ) | tn ( y ) − ¯ f < − ǫ (cid:9) ≤ X n ≥ m t X y ∈ σ − n ( x ) { S n f ( y ) ≤ t } ( y ) ≤ e st (cid:16) e m t P ( − sf ) − e P ( − sf ) (cid:0) H s + R s (cid:1)(cid:17) . ≤ D e st + m t P ( − sf ) where D = D ( s, f, ǫ ) = H s + R s − e P ( − sf ) .Since dds (cid:12)(cid:12) s = δ P ( − sf ) = − f < s > δ so that b := s + 1 + ǫ ¯ f P ( − sf ) < δ. Notice that b does depend on ǫ .With this choice of s , (cid:8) y ∈ W ( x, t ) | tn ( y ) − ¯ f < − ǫ (cid:9) ≤ D e b t . One can similarly show that there exist D > b ∈ ( d ( f ) , δ ) so that (cid:8) y ∈ W ( x, t ) | tn ( y ) − ¯ f > ǫ (cid:9) ≤ D e b t . (In this case, we choose r ∈ ( d ( f ) , δ ) so that b := r + 1 − ǫ ¯ f P ( − rf ) < δ where ǫ = ǫf + ǫ >
0. We then use Equation (5.2) and an analysis similar to the one above to showthat (cid:8) y ∈ W ( x, t ) | tn ( y ) − ¯ f > ǫ (cid:9) ≤ D e t (cid:0) r + − ǫ f P ( − rf ) (cid:1) where D = D ( r, f, ǫ ) = e P ( − rf ) ( H r + R r ).)So, W ( x, t, > ǫ ) ≤ D e b t + D e b t l ≤ De bt where D = D + D and b = max { b , b } < δ . (cid:3) Corollary 5.3.
Suppose that Σ + is a topologically mixing, one-sided, countable Markov shift with(BIP) and f : Σ + → R is a strictly positive, locally H¨older continuous function with a weak entropygap at infinity. Let δ > d ( f ) be the unique constant such that P ( − δf ) = 0 . Then, given any ǫ > ,there exists a > so that (1) There exists ˆ D > so that W ( x, t, > ǫ ) W ( x, t ) ≤ ˆ De − at for any non-periodic x ∈ Σ + . OUNTING AND EQUIDISTRIBUTION 20 (2)
Given any cylinder p , there exists D p so that W ( x, t, > ǫ ) ∩ p ) W ( x, t ) ∩ p ) ≤ D p e − at for any non-periodic x ∈ Σ + .Proof. By Corollary 4.4 we can apply the Renewal Theorem with φ = to see that(5.3) N f ( , x, t ) = W ( x, t ) + 1 = X n ≥ X σ n ( y )= x { S n f ( y ) ≤ t } ( y ) ∼ h − δf ( x ) δ ¯ f e tδ uniformly in x ∈ Σ + , where ∼ indicates that the ratio goes to 1 as t → ∞ . Since there exist b < δ and D > W ( x, t, > ǫ ) ≤ De bt , (1) holds with a = δ − b and some ˆ D > φ = p to conclude that N f ( p , x, t ) = W ( x, t ) ∩ p ) + 1 = X n ≥ X σ n ( y )= x p { S n f ( y ) ≤ t } ( y ) ∼ ν ( p ) h − δf ( x ) δ ¯ f e tδ uniformly in x ∈ Σ + . Since ν ( p ) > W ( x, t, > ǫ ) ∩ p ) ≤ W ( x, t, > ǫ ) ≤ De bt , (2) holds for some D p depending on the cylinder p . (cid:3) The following result will allow us to bound the error terms in our approximations. Given
T > P kT = { p ∈ Λ k | S k f ( z p ) ≤ T } and Q kT = Λ k − P kT . Notice that P kT is finite for all k and T . Corollary 5.4.
Suppose that Σ + is a topologically mixing, one-sided, countable Markov shift with(BIP) and f : Σ + → R is a strictly positive, locally H¨older continuous function with a weak entropygap at infinity. Let δ > d ( f ) be the unique constant such that P ( − δf ) = 0 . (1) There exists
G > so that X n ≥ X { y ∈ σ − n ( x ) } n { S n f ( y ) ≤ t } ( y ) ≤ G e tδ t for any x ∈ Σ + and all t > . (2) If k ∈ N and t > T > , then X n>k X { y ∈ σ − n ( x ) } n Q kT ( y ) { S n f ( y ) ≤ t } ( y ) ≤ Ge − T δ e tδ t − T .
Proof.
Fix some ǫ >
0. Recall from Lemma 4.3 that W ( x, t ) ≤ Ce tδ for all x ∈ Σ + . Then X n ≥ X y ∈ σ − n ( x ) n { S n f ( y ) ≤ t } ( y ) = X y ∈W ( x,t, ≤ ǫ ) n ( y ) + X y ∈W ( x,t,>ǫ ) n ( y ) ≤ X y ∈W ( x,t, ≤ ǫ ) (cid:18) ¯ f + ǫt (cid:19) ( y ) + X y ∈W ( x,t,>ǫ ) ( y ) . ≤ Ce tδ (cid:18) ¯ f + ǫt (cid:19) + (cid:16) ˆ De − at (cid:17) Ce tδ . So, (1) holds for some
G > OUNTING AND EQUIDISTRIBUTION 21
Now notice that X n>k X y ∈ σ − n ( x ) n Q kT ( y ) { S n f ( y ) ≤ t } ( y ) ≤ X n>k n X y ∈ σ k − n ( x ) { S n − k f ( y ) ≤ t − T } ( y )= X m ≥ X w ∈ σ − m ( x ) m + k { S m f ( w ) ≤ t − T } ( w ) ≤ X m ≥ X w ∈ σ − m ( x ) m { S m f ( w ) ≤ t − T } ( w ) ≤ Ge − δT e tδ t − T which completes the proof of (2). (cid:3) Counting
Proof of Theorem A.
First notice that Lemma 3.2 implies that we may assume that f is strictlypositive and has a weak entropy gap at infinity. We simplify notation by setting µ = µ − δf , ν = ν − δf , h = h − δf , and ¯ f = R f dµ .Suppose that p ∈ Λ k . Corollary 4.4 implies that we can apply the Renewal Theorem (Theorem4.1) with φ = p . Therefore, L ( p, t ) := W ( z p , t ) ∩ p ) = X n ≥ X y ∈ σ − n ( z p ) p ( y ) { S n f ( y ) ≤ t } ( y ) ∼ C ( p ) e tδ where C ( p ) = h ( z p ) ν ( p ) δ ¯ f . Fix, for the moment, p ∈ Λ k . We define b L ( p, t ) := X n ≥ n W ( n, p, t ) = X y ∈W ( z p ,t ) n ( y ) p ( y ) . Then b L ( p, t ) = X y ∈W ( z p ,t, ≤ ǫ ) n ( y ) p ( y ) + X y ∈W ( z p ,t,>ǫ ) n ( y ) p ( y ) ≤ X y ∈W ( z p ,t, ≤ ǫ ) (cid:18) ¯ f + ǫt (cid:19) p ( y ) + X y ∈W ( z p ,t,>ǫ ) p ( y ) . Since, by Corollary 5.3, (cid:16) W ( z p , t, > ǫ ) ∩ p (cid:17) ≤ D p e − at (cid:16) W ( z p , t ) ∩ p (cid:17) for some D p , a >
0, it follows that lim sup t →∞ t b L ( p, t ) L ( p, t ) ≤ ¯ f + ǫ. Similarly, b L ( p, t ) = X n ≥ n W ( n, p, t ) ≥ X y ∈W ( z p ,t, ≤ ǫ ) (cid:18) ¯ f − ǫt (cid:19) p ( y ) OUNTING AND EQUIDISTRIBUTION 22 so lim inf t →∞ t b L ( p, t ) L ( p, t ) ≥ ¯ f − ǫ. By letting ǫ → , we see that b L ( p, t ) ∼ ¯ f L ( p, t ) t ∼ C ( p ) ¯ ft e tδ . Now suppose that P is a subset of Λ k and define L ( P, t ) = X p ∈ P L ( p, t ) and b L ( P, t ) = X p ∈ P b L ( p, t ) . The above analysis implies that if P is finite , then L ( P, t ) ∼ X p ∈ P C ( p ) e tδ and b L ( P, t ) ∼ X p ∈ P C ( p ) ¯ ft e tδ . Notice that if
T > t > T , then Corollary 5.4 and Lemma 5.1 imply that there exists C k > t b L ( P kT , t ) e tδ ≤ t b L (Λ k , t ) e tδ ≤ t b L ( P kT , t ) e tδ + tC k e ( s − δ ) t + Ge − δT tt − T for some s ∈ ( d ( f ) , δ ), so¯ f X p ∈ P kT C ( p ) ≤ lim inf t →∞ t b L (Λ k , t ) e tδ ≤ lim sup t →∞ t b L (Λ k , t ) e tδ ≤ ¯ f X p ∈ P kT C ( p ) + Ge − δT Applying the above inequality to the sequence { P kT } T ∈ N , we conclude that b L (Λ k , t ) ∼ X p ∈ Λ k C ( p ) ¯ ft e tδ . Lemma 5.1 implies that, given k ∈ N there exists s < δ and C k >
0, so that X p ∈ Λ k k X n =1 n W ( n, p, t ) ≤ C k e st and k X n =1 n (cid:0) M f ( n, t ) (cid:1) ≤ C k e st and X p ∈ Λ k ∞ X n = k n W ( n, p, t − ǫ k ) ≤ ∞ X n = k n (cid:0) M f ( n, t ) (cid:1) ≤ X p ∈ Λ k ∞ X n = k n W ( n, p, t + ǫ k ) . Therefore, recalling that M f ( t ) = P n ≥ n (cid:0) M f ( n, t ) (cid:1) , we see that b L (Λ k , t − ǫ k ) − C k e st ≤ M f ( t ) ≤ b L (Λ k , t + ǫ k ) + C k e st , so e − δǫ k ¯ f X p ∈ Λ k C ( p ) ≤ lim inf t →∞ tM f ( t ) e tδ ≤ lim sup t →∞ tM f ( t ) e tδ ≤ e δǫ k ¯ f X p ∈ Λ k C ( p )Since h is bounded and continuous and v k = sup { µ ( p ) | p ∈ Λ k } → k → ∞ , by Lemma 5.1(i), X p ∈ Λ k C ( p ) = 1 δ ¯ f X p ∈ Λ k h ( z p ) ν ( p ) → R h dνδ ¯ f = 1 δ ¯ f . as k → ∞ . Moreover, lim ǫ k = 0. So, finally, we may conclude that M f ( t ) ∼ e tδ tδ OUNTING AND EQUIDISTRIBUTION 23 as desired. (cid:3) Equidistribution
We are almost ready to prove our equidistribution result, but first we must develop one morebound in the spirit of [33, Theorem 6].7.1.
Preparing to equidistribute.
Suppose that f : Σ + → R and g : Σ + → R are both strictlypositive, f has a weak entropy gap at infinity and P ( − δf ) = 0. We simplify notation, throughoutthe section, by letting µ = µ − δf denote the equilibrium state of − δf and setting f := R f dµ and g := R g dµ . Since f and g are strictly positive, c ( f ) = inf { f ( x ) | x ∈ Σ + } > c ( g ) = inf { g ( x ) | x ∈ Σ + } > . Proposition 7.1.
Suppose that Σ + is a topologically mixing, one-sided, countable Markov shiftwith (BIP) and f : Σ + → R is a strictly positive, locally H¨older continuous function with a weakentropy gap at infinity. Let δ > d ( f ) be the unique constant such that P ( − δf ) = 0 . Further supposethat g : Σ + → R is strictly positive and that there exists C > so that | f ( x ) − g ( x ) | ≤ C for all x ∈ Σ + . Given ǫ > , there exist A > and a < δ so that n y ∈ W ( x, t ) : (cid:12)(cid:12)(cid:12) S n g ( y ) n ( y ) − ¯ g (cid:12)(cid:12)(cid:12) > ǫ, (cid:12)(cid:12) tn ( y ) − f (cid:12)(cid:12) ≤ ǫ o ≤ Ae − at for any non-periodic x ∈ Σ + .Proof. Fix ǫ >
0. We may assume that ǫ < min { c ( f ) , c ( g ) } .If S n ( y ) g ( y ) n ( y ) − g < − ǫ , then S n ( y ) g ( y ) < n ( y ) g − n ( y ) ǫ. If, in addition, (cid:12)(cid:12) tn ( y ) − f (cid:12)(cid:12) ≤ ǫ , then t ≤ n ( y )( f + ǫ ), so S n ( y ) g ( y ) < n ( y ) g − n ( y ) ǫ ≤ n ( y ) g − n ( y ) ǫ − tǫ f + ǫ ) ≤ n ( y )( g − ǫ ) − tǫ where ǫ = max { ǫ , ǫ f + ǫ ) } > s → P ( − sg − δf ) is monotone decreasing and well-defined on( d ( f ) − δ, ∞ ). So, if s >
0, then P ( − sg − δf ) <
0. Moreover, there exist an equilibrium state µ − sg − δf for − sg − δf and an eigenfunction h − sg − δf for L − sg − δf with eigenvalue e P ( − sg − δf ) < dds (cid:12)(cid:12) s =0 P ( − sg − δf ) = − g < s > − d := s ( g − ǫ ) + P ( − sg − δf ) < . Theorem 2.6 implies that there exist ¯ R s > η s ∈ (0 ,
1) so that(7.1) (cid:13)(cid:13)(cid:13) e − nP ( − sg − δf ) L − sg − δf − h − sg − δf ( x ) Z dν − sg − δf (cid:13)(cid:13)(cid:13) ≤ R s ¯ η ns for all n ∈ N . Therefore, n y ∈ W ( x, t ) : S n g ( y ) n ( y ) − ¯ g < − ǫ, (cid:12)(cid:12) tn ( y ) − f (cid:12)(cid:12) ≤ ǫ o ≤ X n ≥ X σ n ( y )= x { y | S n g ( y ) ≤ n · ( g − ǫ ) − tǫ , S n f ( y ) ≤ t } ( y ) ≤ X n ≥ X σ n ( y )= x e − s (cid:0) S n g ( y ) − n ( g − ǫ )+ tǫ (cid:1) − δ (cid:0) S n f ( y ) − t (cid:1) = e tδ − stǫ X n ≥ e n ( s (¯ g − ǫ )+ P ( − sg − δf )) (cid:16) e − nP ( − sg − δf ) L n − sg − δf (cid:17) ≤ e tδ − stǫ X n ≥ (cid:0) h − sg − δf ( x ) + ¯ R s ¯ η ns (cid:1) e − nd ≤ D e tδ − stǫ OUNTING AND EQUIDISTRIBUTION 24 for all x ∈ Σ + , and some D > ǫ , s , g and f ).One may similarly show that there exist ǫ > r < D > n y ∈ W ( x, t ) : S n g ( y ) n ( y ) − ¯ g > ǫ, (cid:12)(cid:12) tn ( y ) − f (cid:12)(cid:12) ≤ ǫ o ≤ D e tδ + rtǫ . Therefore, our result holds with A = D + D and a = max { δ − sǫ , δ + rǫ } . (cid:3) Proof of Theorem B.
Lemma 3.2 again implies that we may assume that f and g are strictlypositive and f has a weak entropy gap at infinity. Recall, from Lemma 5.1, that there exists asequence { ǫ k } so that lim ǫ k = 0, and, for any p ∈ Λ k and n ≥ k , there exists a bijectionΨ np : Fix n ∩ p → σ − n ( z p ) ∩ p so that | S n f ( x ) − S n f (Ψ np ( x )) | ≤ ǫ k and | S n g ( x ) − S n g (Ψ np ( x )) | ≤ ǫ k for all x ∈ Fix n ∩ p . Since lim ǫ k = 0, there exists k so that if n ≥ k ≥ k , then c = min { c ( f ) , c ( g ) } > ǫ k . We assume from now on that k ≥ k . Then, if p ∈ Λ k (7.2) X n ≥ k n X x ∈ Fix n ∩ pSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ X y ∈W ( z p ,t + ǫ k ) ∩ p n ( y ) (cid:18) S n g ( y ) + ǫ k S n f ( y ) − ǫ k (cid:19) { n ( y ) ≥ k } ( y )and(7.3) X n ≥ k n X x ∈ Fix n ∩ pSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≥ X y ∈W ( z p ,t − ǫ k ) ∩ p n ( y ) (cid:18) S n g ( y ) − ǫ k S n f ( y ) + ǫ k (cid:19) { n ( y ) ≥ k } ( y ) . Since there exists
C > | f ( x ) − g ( x ) | ≤ C for all x ∈ Σ + , S n ( y ) f ( y ) ≥ cn ( y ) for all y ∈ Σ + and c > ǫ k , we see that S n g ( y ) S n f ( y ) ≤ nC + S n f ( y ) S n f ( y ) ≤ ˆ C = Cc + 1 and S n ( y ) g ( y ) + ǫ k S n ( y ) f ( y ) − ǫ k ≤ C. Let V ( x, t, ≤ ǫ ) = n y ∈ W ( x, t ) : (cid:12)(cid:12)(cid:12) S n f ( y ) n ( y ) − ¯ f (cid:12)(cid:12)(cid:12) ≤ ǫ, (cid:12)(cid:12)(cid:12) S n g ( y ) n ( y ) − ¯ g (cid:12)(cid:12)(cid:12) ≤ ǫ o . Given ǫ > ǫ + 2 ǫ k < ¯ f . Proposition 5.2 together with Proposition 7.1, applied to both f and g , imply that there exist ˆ A > a < δ so that (cid:0) W ( x, t ) \ V ( x, t, ≤ ǫ ) (cid:1) ≤ ˆ Ae ˆ at for all t >
0. Further recall that we saw in the proof of Theorem A that b L ( p, t ) = X y ∈W ( z p ,t + ǫ k ) ∩ p n ( y ) ∼ C ( p ) ¯ f e tδ t . OUNTING AND EQUIDISTRIBUTION 25
Notice that U ( p, t + ǫ k ) := X y ∈W ( z p ,t + ǫ k ) ∩ p n ( y ) (cid:18) S n ( y ) g ( y ) + ǫ k S n ( y ) f ( y ) − ǫ k (cid:19) ≤ X y ∈V ( z p ,t + ǫ k , ≤ ǫ ) ∩ p n ( y ) g + ǫ + ǫ k n ( y ) f − ǫ − ǫ k n ( y ) ! + 3 ˆ C (cid:16) W ( z p , t + ǫ k ) \ V ( z p , t + ǫ k , ≤ ǫ ) (cid:17) +3 ˆ C k − X n =1 W ( n, p, t )and recall, from Lemma 5.1, that given s ∈ ( d ( f ) , δ ), there exists C ( k, s ) so that W ( n, p, t ) ≤ C ( k, s ) e st and M f ( t ) ≤ C ( k, s ) e st for all n < k . Therefore, lim sup t →∞ U ( p, t + ǫ k )ˆ L ( p, t + ǫ k ) ≤ g + ǫ + ǫ k f − ǫ − ǫ k . Letting ǫ →
0, we see that lim sup t →∞ U ( p, t + ǫ k )ˆ L ( p, t + ǫ k ) ≤ g + ǫ k f − ǫ k . We can similarly show that if Z ( p, t − ǫ k ) = X y ∈W ( z p ,t − ǫ k ) n ( y ) p ( y ) (cid:18) S n g ( y ) − ǫ k S n f ( y ) + ǫ k (cid:19) , then lim inf t →∞ Z ( p, t − ǫ k )ˆ L ( p, t − ǫ k ) ≥ g − ǫ k f + ǫ k . Therefore, g − ǫ k f + ǫ k ≤ lim inf t →∞ b L ( p, t − ǫ k ) X n ≥ k n X x ∈ Fix n ∩ pSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ lim sup t →∞ b L ( p, t + ǫ k ) X n ≥ k n X x ∈ Fix n ∩ pSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ g + ǫ k f − ǫ k . Since P kT is a finite set of cylinders, for any T and k , we see that g − ǫ k f + ǫ k ≤ lim inf t →∞ b L ( P T , t − ǫ k ) X n ≥ k n X x ∈ Fix n ∩ PkTSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ lim sup t →∞ b L ( P kT , t + ǫ k ) X n ≥ k n X x ∈ Fix n ∩ PkTSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ g + ǫ k f − ǫ k . Now notice that if t > T >
0, Corollary 5.4 implies that(7.4) X n ≥ k n X x ∈ Fix n ∩ QkTSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ C b L ( Q kT , t ) ≤ CGe − δT e tδ t − T Therefore, as in the proof of Theorem A, we conclude that g − ǫ k f + ǫ k ≤ lim inf t →∞ b L (Λ k , t − ǫ k ) X n ≥ k n X x ∈ Fix nSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ lim sup t →∞ b L (Λ k , t + ǫ k ) X n ≥ k n X x ∈ Fix nSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ≤ g + ǫ k f − ǫ k . Recall that lim ǫ k = 0, b L (Λ k , t − ǫ k ) − C k e st ≤ M f ( t ) ≤ b L (Λ k , t + ǫ k ) + C k e st , OUNTING AND EQUIDISTRIBUTION 26 for all t >
0, and that lim t →∞ M f ( t ) tδe δt = 1 , so we see that ∞ X n =1 n X x ∈ Fix nSnf ( x ) ≤ t S n g ( x ) S n f ( x ) ∼ ¯ g ¯ f e tδ tδ as desired. This completes the proof of Theorem B.8. The Manhattan curve
Suppose that f : Σ + → R is locally H¨older continuous, strictly positive and has a strong entropygap at infinity and that g : Σ + → R is also strictly positive and locally H¨older continuous and thereexists C > | f ( x ) − g ( x ) | < C for all x ∈ Σ + . In this case, c ( f ) = inf { f ( x ) | x ∈ Σ + } > c ( g ) = inf { g ( x ) | x ∈ Σ + } > Manhattan curve C ( f, g ) = { ( a, b ) ∈ D ( f, g ) | P ( − af − bg ) = 0 } where D ( f, g ) = (cid:8) ( a, b ) ∈ R | ac ( f ) + bc ( g ) > a + b > (cid:9) . Notice that if f : Σ + → R and g : Σ + → R are both eventually positive and locally H¨oldercontinuous, f has a strong entropy gap at infinity and there exists C so that | f ( x ) − g ( x ) | ≤ C for all x ∈ Σ + , Lemma 3.2 implies that f and g are cohomologous to ˆ f : Σ → R and ˆ g : Σ + → R (respectively) which are both strictly positive and locally H¨older continuous, ˆ f has a strong entropygap at infinity and there exists ˆ C so that | ˆ f ( x ) − ˆ g ( x ) | ≤ C for all x ∈ Σ + . Since C ( f, g ) = C ( ˆ f , ˆ g ),Theorem C follows from the following stronger statement for strictly positive functions. Theorem C*:
Suppose that (Σ + , σ ) is a topologically mixing, one-sided countable Markov shiftwith (BIP), f : Σ + → R is locally H¨older continuous, strictly positive and has a strong entropy gapat infinity and g : Σ + → R is also strictly positive and locally H¨older continuous. If there exists C > so that | f ( x ) − g ( x ) | < C for all x ∈ Σ + , then (1) ( δ ( f ) , , (0 , δ ( g )) ∈ C ( f, g ) . (2) If ( a, b ) ∈ D ( f, g ) , there exists a unique t > d ( f ) a + b so that ( ta, tb ) ∈ C ( f, g ) . (3) C ( f, g ) is an analytic curve. (4) C ( f, g ) is strictly convex, unless (8.1) S n f ( x ) = δ ( g ) δ ( f ) S n g ( x ) for all x ∈ Fix n and n ∈ N .Moreover, the tangent line to C ( f, g ) at ( a, b ) has slope s ( a, b ) = − R Σ + g dµ − af − bg R Σ + f dµ − af − bg . Proof.
By definition, ( δ ( f ) ,
0) and (0 , δ ( g )) lie on C ( f, g ) so (1) holds.Notice that, since (cid:12)(cid:12) I ( f, a ) − I ( g, a ) (cid:12)(cid:12) ≤ C for all a ∈ A , d ( f ) = d ( g ) and g also has a strongentropy gap at infinity. Moreover, if ( a, b ) ∈ D ( f, g ), then af + bg is strictly positive, has a strongentropy gap at infinity and d ( af + bg ) = d ( f ) a + b . OUNTING AND EQUIDISTRIBUTION 27
Lemma 3.3 then implies that if ( a, b ) ∈ D ( f, g ), then t → P ( − t ( af + bg )) is proper and strictlydecreasing on ( d ( f ) a + b , ∞ ) so there exists a unique t > d ( f ) a + b so that P ( − t ( af + bg )) = 0. Thus, (2)holds.Lemma 3.4 implies that there is an equilibrium state µ − af − bg for − af − bg and that R Σ + ( − af − bg ) dµ − af − bg is finite. Notice that if ( c, d ) ∈ D ( f, g ), then the ratio cf + dgaf + bg is bounded, this implies that R Σ + ( cf + dg ) dµ − af − bg is also finite. Theorem 2.4 then implies that if ( a, b ) ∈ D ( f, g ), then ∂∂a P ( − af − bg ) = Z Σ + − f dµ − af − bg and ∂∂b P ( − af − bg ) = Z Σ + − g dµ − af − bg . Since f is strictly positive, R Σ + − f dµ − af − bg is non-zero, so P is a submersion on D ( f, g ). Theimplicit function theorem then implies that C ( f, g ) = { ( a, b ) ∈ D ( f, g ) | P ( − af − bg ) = 0 } is an analytic curve and that if ( a, b ) ∈ C ( f, g ) then the slope of the tangent line to C ( f, g ) at( a, b ) is given by s ( a, b ) = − R Σ + g dµ − af − bg R Σ + f dµ − af − bg . Since P is convex, see Sarig [59, Proposition 4.4], C ( f, g ) is convex. A convex analytic curve isstrictly convex if and only if it is not a line. So it remains to show that f and g satisfy equation(8.1) if and only if C ( f, g ) is a straight line.If C ( f, g ) is a straight line, then by (1) it has slope − δ ( f ) δ ( g ) . In particular,(8.2) − s ( δ ( f ) ,
0) = δ ( f ) δ ( g ) = R Σ + g dµ − δ ( f ) f R Σ + f dµ − δ ( f ) f = R Σ + g dµ − δ ( g ) g R Σ + f dµ − δ ( g ) g . By definition, h σ ( µ − δ ( g ) g ) − δ ( g ) Z Σ + g dµ − δ ( g ) g = 0so, applying equation (8.2), we see that h σ ( µ − δ ( g ) g ) − δ ( f ) Z Σ + f dµ − δ ( g ) g = δ ( g ) Z Σ + g dµ − δ ( g ) g − δ ( f ) Z Σ + f dµ − δ ( g ) g = 0Since P ( − δ ( f ) f ) = 0, this implies that µ − δ ( g ) g is an equilibrium state for − δ ( f ) f . Therefore, byuniqueness of equilibrium states we see that µ − δ ( f ) f = µ − δ ( g ) g . Sarig [59, Thm. 4.8] showed thatthis only happens when − δ ( f ) f and − δ ( g ) g are cohomologous, so the Livsic Theorem (Theorem2.1) implies that this occurs if and only if S n f ( x ) = δ ( g ) δ ( f ) S n g ( x )for all x ∈ Fix n and n ∈ N . We have completed the proof. (cid:3) Background for applications
In this section, we recall the background material that we will need to construct the roof func-tions described in Theorem D. We will also recall the more general definition of cusped Anosovrepresentations of geometrically finite Fuchsian groups into SL ( d, R ). In the next section, we willsee that Theorem D also extends to this setting. OUNTING AND EQUIDISTRIBUTION 28
Linear algebra.
It will be useful to first recall some standard Lie-theoretic notation. Let a = { ( a , . . . , a d ) ∈ R d | a + . . . + a d = 0 } be the standard Cartan algebra for SL ( d, R ) and let a + = { ( a , . . . , a d ) ∈ a | a ≥ · · · ≥ a d } be the standard choice of positive Weyl chamber. Let a ∗ be the space of linear functionals on a .For all k ∈ { , . . . , d − } , let α k : a → R be given by α k ( ~a ) = a k − a k +1 . Then { α , . . . , α d − } span a ∗ and are the simple roots determining the Weyl chamber a + . It is also natural to consider thefundamental weights ω k ∈ a ∗ given by ω k ( ~a ) = a + · · · + a k . Notice that { ω , . . . , ω d − } is also abasis for a ∗ .If A ∈ SL ( d, R ), let λ ( A ) ≥ λ ( A ) ≥ · · · ≥ λ d ( A )denote the moduli of the generalized eigenvalues of A and let σ ( A ) ≥ σ ( A ) ≥ · · · ≥ σ d ( A )be the singular values of A . The Jordan projection ℓ : SL ( d, R ) → a + is given by ℓ ( A ) = (log λ ( A ) , . . . , log λ d ( A ))and the Cartan projection κ : SL ( d, R ) → a + is given by κ ( A ) = (log σ ( A ) , . . . , log σ d ( A )) . If α k ( ℓ ( A )) >
0, then there is a well-defined attracting k -plane which is the plane spanned bythe generalized eigenspaces with eigenvalues of modulus at least λ k ( A ). Recall that the Cartandecomposition of A ∈ SL ( d, R ) has the form A = KDL where
K, L ∈ SO ( d ) and D is the diagonalmatrix with diagonal entries d ii = σ i ( A ). If α k ( A ) >
0, then the k -flag U k ( A ) = K (cid:0) h e , . . . , e k i (cid:1) is well-defined, and is the k -plane spanned by the k longest axes of the ellipsoid A ( S d − ). (Noticethat U k ( A ) is not typically the attracting k -plane even when α k ( ℓ ( A )) > Cusped Anosov representations of geometrically finite Fuchsian groups.
Supposethat Γ ⊂ PSL (2 , R ) is a torsion-free geometrically finite Fuchsian group, which is not convexcocompact, and let Λ(Γ) be its limit set in ∂ H .Consider a representation ρ : Γ → SL ( d, R ) such that if α ∈ Γ is parabolic, then ρ ( α ) is unipotent.If 1 ≤ k ≤ d −
1, we will say that ρ is cusped P k -Anosov if there exist continuous ρ -equivariantmaps ξ kρ : Λ(Γ) → Gr k ( R d ) and ξ d − kρ : Λ(Γ) → Gr d − k ( R d ) so that(1) ξ kρ and ξ d − kρ are transverse , i.e. if x = y ∈ Λ(Γ), then ξ kρ ( x ) ⊕ ξ d − kρ ( y ) = R d . (2) ξ kρ and ξ d − kρ are strongly dynamics preserving , i.e. if j is k or d − k and { γ n } is a sequencein Γ so that γ n (0) → x ∈ Λ(Γ) and γ − n (0) → y ∈ Λ(Γ), then if V ∈ Gr j ( R d ) and V istransverse to ξ d − jρ ( y ), then ρ ( γ n )( V ) → ξ jρ ( x ).The original definition of a cusped P k -Anosov representation in [13] is given in terms of a flowspace, as in Labourie’s original definition [31]. The characterization we give here is a naturalgeneralization of characterizations of Gu´eritaud-Guichard-Kassel-Wienhard [21], Kapovich-Leeb-Porti [28] and Tsouvalas [64] in the traditional setting. Our cusped P k -Anosov representationsare examples of the relatively Anosov representations considered by Kapovich-Leeb [27] and therelatively dominated representations considered by Zhu [66].The following crucial properties of cusped P k -Anosov representations are established in Canary-Zhang-Zimmer [13]. (Several of these properties also follow from work of Kapovich-Leeb [27] and OUNTING AND EQUIDISTRIBUTION 29
Zhu [66] once one establishes that our representations fit into their framework.) If ρ : Γ → SL ( d, R )is cusped P k -Anosov, we define the space of type-preserving deformationsHom tp ( ρ ) ⊂ Hom(Γ , SL ( d, R ))to be the space of representations σ such that if α ∈ Γ is parabolic, then σ ( α ) is conjugate to ρ ( α ). Theorem 9.1. (Canary-Zhang-Zimmer [13]) If Γ is a geometrically finite Fuchsian group and ρ : Γ → SL ( d, R ) is a cusped P k -Anosov representation, then (1) There exist
A, a > so that if γ ∈ Γ , then Ae ad ( b ,γ ( b )) ≥ e α k ( κ ( ρ ( γ ))) ≥ A e d ( b ,γ ( b a where b is a basepoint for H . (2) There exist
B, b > so that if γ ∈ Γ , then Be bt ( γ ) ≥ e α k ( ℓ ( ρ ( γ ))) ≥ B e t ( γ ) b where t ( γ ) is the translation length of γ on H . (3) The limit maps ξ kρ and ξ d − kρ are H¨older continuous. (4) There exists an open neighborhood U of ρ in Hom tp ( ρ ) , so that if σ ∈ U , then σ is cusped P k -Anosov. (5) If υ ∈ Γ is parabolic and j ∈ { , . . . , d − } , then there exists c j ( ρ, υ ) ∈ Z and C j ( ρ, υ ) > so that (cid:12)(cid:12) α j ( κ ( ρ ( υ n ))) − c j ( ρ, υ ) log n (cid:12)(cid:12) < C j ( ρ, υ ) for all n ∈ N . Moreover, if η ∈ Hom tp ( ρ ) , then c j ( ρ, υ ) = c j ( η, υ ) . (6) ρ has the P k -Cartan property , i.e. whenever { γ n } is a sequence of distinct elements of Γ such that γ n ( b ) converges to z ∈ Λ(Γ) , then ξ kρ ( z ) = lim U k ( ρ ( γ n )) . (7) ρ is P d − k -Anosov. Cusped Hitchin representations.
Canary, Zhang and Zimmer [13] also prove that cuspedHitchin representations are cusped P k -Anosov for all k , i.e they are cusped Borel Anosov, in analogywith work of Labourie [31] in the uncusped case. We say that A ∈ SL ( d, R ) is unipotent and totallypositive with respect to a basis b = ( b , . . . , b d ) for R d , if its matrix representative with respect tothis basis is unipotent, upper triangular, and all the minors which could be positive are positive.Let U > ( b ) denote the set of all such maps. One crucial property here is that U > ( b ) is a semi-group(see Lusztig [38]).We say that a basis b = ( b , . . . , b d ) is consistent with a pair ( F, G ) of transverse flags if h b i i = F i ∩ G d − i +1 for all i . A k -tuple ( F , . . . , F k ) in F d is positive if there exists a basis b consistent with( F , F k ) and there exists { u , . . . , u d } ∈ U ( b ) > so that F i = u i · · · u F for all i = 2 , . . . , d .If X is a subset of S , we say that a map ξ : X → F d is positive if whenever ( x , . . . , x k ) is a consis-tently ordered k -tuple in X (ordered either clockwise or counter-clockwise), then ( ξ ( x ) , . . . , ξ ( x k ))is a positive k -tuple of flags.A cusped Hitchin representation is a representation ρ : Γ → SL ( d, R ) such that if γ ∈ Γ is para-bolic, then ρ ( γ ) is a unipotent element with a single Jordan block and there exists a ρ -equivariantpositive map ξ ρ : Λ(Γ) → F d . (In fact, it suffices to define ξ ρ on the subset Λ per (Γ) consisting offixed points of peripheral elements of Γ.) Theorem 9.2. (Canary-Zhang-Zimmer [13]) If Γ is a geometrically finite Fuchsian group and ρ : Γ → SL ( d, R ) is a cusped Hitchin representation, then (1) ρ is P k -Anosov for all ≤ k ≤ d − . (2) ρ is irreducible. (3) If α ∈ Γ is parabolic and ≤ k ≤ d − , then c k ( ρ, α ) = 2 . OUNTING AND EQUIDISTRIBUTION 30
We remark that Sambarino [55] has independently established that ρ is irreducible and thatKapovich-Leeb indicate in [27] that they can prove ρ is Borel Anosov.9.4. Codings for geometrically finite Fuchsian groups.
A torsion-free convex cocompactFuchsian group admits a finite Markov shift which codes the recurrent portion of its geodesic flow.The most basic such coding is the Bowen-Series coding [7]. However, if the group is geometricallyfinite, but not convex cocompact, this coding is not well-behaved. In this case one must insteadconsider the countable Markov shifts constructed by Dal’bo-Peign´e [19], if the quotient has infinitearea, and Stadlbauer [62] and Ledrappier-Sarig [35], if the quotient has finite area.We summarize the crucial properties of these Markov shifts in the following theorem and willgive a brief description of each coding.
Theorem 9.3. (Dal’bo-Peign´e [19], Ledrappier-Sarig [35], Stadlbauer [62])
Suppose that Γ is atorsion-free geometrically finite, but not convex cocompact, Fuchsian group. There exists a topolog-ically mixing Markov shift (Σ + , A ) with countable alphabet A with (BIP) which codes the recurrentportion of the geodesic flow U Γ . There exist maps G : A → Γ , ω : Σ + → Λ(Γ) , r : A → N , and s : A → Γ with the following properties. (1) ω is locally H¨older continuous, injective and ω (Σ + ) = Λ c (Γ) , i.e. the complement in Λ(Γ) of the set of fixed points of parabolic elements of Γ . Moreover, ω ( x ) = G ( x ) ω ( σ ( x )) forevery x ∈ Σ + . (2) If x ∈ Fix n , then ω ( x ) is the attracting fixed point of G ( x ) · · · G ( x n ) . Moreover, if γ ∈ Γ ishyperbolic, then there exists x ∈ Fix n (for some n ) so that γ is conjugate to G ( x ) · · · G ( x n ) and x is unique up to shift. (3) There exists Q ∈ N such that ≤ r − ( n )) ≤ Q for all n ∈ N . (4) There exists a finite collection P of parabolic elements of Γ , a finite collection R of elementsof Γ such that if a ∈ A , then s ( a ) ∈ P ∪ { id } and G ( a ) = s ( a ) r ( a ) − g a where g a ∈ R . (5) Given a basepoint b ∈ H , there exists L > so that if x ∈ Σ + and n ∈ N , then d (cid:0) G ( x ) · · · G ( x n )( b ) , −−−−→ b ω ( x ) (cid:1) ≤ L. If Γ is convex cocompact, then one may use the Bowen-Series [7] coding (Σ + , σ ) which we brieflyrecall to set the scene for the more complicated codings we will need in the non-convex cocompactsetting. One begins with a fundamental domain D for Γ, containing the basepoint b , all of whosevertices lie in ∂ H , so that the set of face pairings A of D is a minimal symmetric generating setfor Γ. The classical Bowen-Series coding on the alphabet A can be constructed from a “cuttingsequence” which records the intersections ( t k ) of a geodesic ray ←→ b z which intersects D , where z ∈ Λ(Γ), with edges of translates of D so that the geodesic is entering γ k ( D ) as it passesthrough t k . The classical Bowen-Series coding for ←→ b z is given by ( x k ) = ( γ k γ − k − ). Each γ k γ − k +1 is a face-pairing, hence this alphabet A is a finite generating set for Γ. Thus one obtains a map G : A →
Γ, the map ω simply takes the word encoding the geodesic ray −→ b z to z . Moreover, r ( a ) = 1 and s ( a ) = id for all a ∈ A . A word x in A is allowable in this coding if and only if G ( x i +1 ) / ∈ { G ( x i ) , G ( x i ) − } for any i .If Γ is geometrically finite and has infinite area quotient, then we may use the Dal’bo-Peign´ecoding [19]. Roughly, the Dal’bo-Peign´e coding coalesces all powers of a parabolic generator in theBowen-Series coding. This alteration allows ω to be locally H¨older continuous. Here we may beginwith fundamental domain D for Γ, containing the origin 0 in the Poincar´e disk model, all of whosevertices lie in ∂ H , so that the set of face pairings A of D is a minimal symmetric generating setfor Γ and such that every parabolic element of Γ is conjugate to an element of A . Let P denote OUNTING AND EQUIDISTRIBUTION 31 the parabolic elements of A . We let A = A ∪ { p n | n ≥ , p ∈ P} . In all cases, G ( a ) = a . If a = p n for some p ∈ P , then r ( a ) = n + 1, s ( a ) = p and g a = p , while ifnot we set r ( a ) = 1, s ( a ) = id and g a = a . A word x in A is allowable in this coding if and onlyif for any i , G ( x i +1 ) / ∈ { G ( x i ) , G ( x i ) − } and if s ( x i ) ∈ P , then s ( x i +1 ) / ∈ { s ( x i ) , s ( x i ) − } . For adiscussion of this coding in our language, see Kao [25].If Γ is geometrically finite and has a finite area quotient then one cannot use the Dal’bo-Peign´ecoding, since there is not a minimal symmetric generating set which contains elements conjugateto every primitive parabolic element of Γ. Stadlbauer [62] and Ledrappier-Sarig [35] constructa (more complicated) coding in this setting which has the same flavor and coarse behavior asthe Dal’bo-Peign´e coding. One begins with a Bowen-Series coding of Γ with alphabet A . Let C denote a set of minimal length conjugates of primitive parabolic elements. They then choosea sufficiently large even number 2 N so that the length of every element of C divides 2 N andlet P be the collection of powers of elements of C of length exactly 2 N . Let A be the set ofall strings ( b , b , . . . , b N ) in A so that b b · · · b N is freely reduced in A and so that neither b b · · · b N or b b · · · b N − lies in P . Let A be the set of all freely reduced strings of the form( b, υ t , υ , · · · , υ k − , c ) where b ∈ A − { υ N } , υ = υ · · · υ N ∈ P , υ i ∈ A for all i , t ∈ N and c ∈ A − { υ k } . Let A = A ∪ A . If a = ( b , b , . . . , b N ) ∈ A , then G ( a ) = b , r ( a ) = 1, s ( a ) = id and g a = b , while if a = ( b, υ t , υ · · · υ k − , c ), then let G ( a ) = υ t − υ · · · υ k − , r ( a ) = t + 1, s ( a ) = υ and g a = υ · · · υ k − . The set of allowable words is defined so that if x ∈ Fix n , then G ( x ) · · · G ( x n )cannot be a parabolic element of Γ. (For a more detailed description see Stadlbauer [62], Ledrappier-Sarig [35] or Bray-Canary-Kao [9].)9.5. Busemann and Iwasawa cocycles.
We will use the Busemann cocycle to define our rooffunctions. We first develop the theory we will need in the simpler case where ρ is cusped P k -Anosovfor all k . This theory will suffice for all our application to cusped Hitchin representations, so onemay ignore the discussion of partial flag varieties and partial Iwasawa cocycles on a first reading.Quint [48] introduced a vector valued smooth cocycle, called the Iwasawa cocycle , B : SL ( d, R ) × F d → a where F d is the space of (complete) flags in R d . Let F denote the standard flag F = ( h e i , h e , e i , . . . , h e , . . . , e d − i ) . We can write any F ∈ F d as F = K ( F ) where K ∈ SO ( d ). If A ∈ SL ( d, R ) and F ∈ F d , the Iwasawadecomposition of AK has the form QZU where Q ∈ SO ( d ), Z is a diagonal matrix with non-negativeentries, and U is unipotent and upper triangular. Then B ( A, F ) = (log z , . . . , log z dd ).One may check that it satisfies the following cocycle property (see Quint [48, Lemma 6.2]): B ( ST, F ) = B ( S, T F ) + B ( T, F ) . If A is loxodromic (i.e. α k ( ℓ ( A )) > k ), then the set of attracting k -planes forms a flag F A , called the attracting flag of A . In this case,(9.1) B ( A, F A ) = ℓ ( A )since if F A = K A ( F ), then AK A is upper triangular and the diagonal entries are the eigenvalueswith their moduli in descending order. (See Lemma 7.5 in Sambarino [52].)The Iwasawa cocycle is also closely related to the singular value decomposition, also known asthe Cartan decomposition. If A is Cartan loxodromic (i.e. α k ( κ ( A )) > k ), then the flag U ( A ) = { U k ( A ) } is well-defined. If W is the involution taking e i to e d − i +1 and A has Cartandecomposition A = KDL , then A − has Cartan decomposition A − = (cid:0) L − W (cid:1) (cid:0) W D − W (cid:1) (cid:0) W K − (cid:1) . OUNTING AND EQUIDISTRIBUTION 32
So if S ( A ) = U ( A − ), one may check that B ( A, S ( A )) = κ ( A ). Moreover, the Cartan decompositionbounds the Iwasawa cocycle, specifically || B ( A, F ) || ≤ || κ ( A ) || (see Benoist-Quint [2, Corollary 8.20]).We will make use of the following close relationship between the Iwasawa cocycle and the Cartanprojection. Lemma 9.4. (Quint [48, Lemma 6.5])
For any ǫ ∈ (0 , , there exists C > so that if A ∈ SL ( d, R ) , F ∈ F d , σ k ( A ) > σ k +1 ( A ) and ∠ (cid:16) F k , U d − k ( A − ) (cid:17) ≥ ǫ , then (cid:12)(cid:12) ω k ( B ( A, F )) − ω k ( κ ( A )) (cid:12)(cid:12) ≤ C. Given a representation ρ : Γ → SL ( d, R ) of a geometrically finite Fuchsian group Γ and a ρ -equivariant map ξ ρ : Λ(Γ) → F d we define its associated Busemann cocycle β ρ : Γ × Λ(Γ) → a by letting β ρ ( γ, x ) = B (cid:0) ρ ( γ ) , ρ ( γ − )( ξ ρ ( x )) (cid:1) . The Busemann cocycle was first defined by Quint [48] and was previously used to powerful effectin the setting of uncusped Hitchin representations by Sambarino [53], Martone-Zhang [40] andPotrie-Sambarino [47].
Lemma 9.5. If ρ : Γ → SL ( d, R ) is a representation of a geometrically finite Fuchsian group Γ and ξ ρ : Λ(Γ) → F d is a ρ -equivariant map, then β ρ satisfies the cocycle property β ρ ( αγ, z ) = β ρ ( α, z ) + β ρ ( γ, α − ( z )) for all α, γ ∈ Γ and z ∈ Λ(Γ) .Moreover, if ρ ( γ ) is loxodromic and ξ ρ ( γ + ) is the attracting flag of ρ ( γ ) , then β ρ ( γ, γ + ) = ℓ ( ρ ( γ )) . Proof.
First notice that β ρ ( αγ, z ) = B (cid:0) ρ ( α ) ρ ( γ ) , ρ ( γ − ) ρ ( α − )( ξ ρ ( z )) (cid:1) = B (cid:0) ρ ( α ) , ρ ( α ) − ( ξ ρ ( z )) (cid:1) + B (cid:0) ρ ( γ ) , ρ ( γ − ) ρ ( α − )( ξ ρ ( z )) (cid:1) = β ρ ( α, z ) + β ρ ( γ, α − ( z )) . Then observe that β ρ ( γ, γ + ) = B (cid:0) ρ ( γ ) , ρ ( γ − )( ξ ρ ( γ + )) (cid:1) = B ( ρ ( γ ) , ξ ρ ( γ + )) . Since we have assumed that ξ ρ ( γ + ) is the attracting flag of ρ ( γ ), we may apply Equation (9.1). (cid:3) We now generalize the theory developed above to the setting of partial flag varieties. If θ = { i < · · · < i r } ⊂ { , . . . , d } , then a θ -flag is a nested collection of vector subspaces of dimension i j of the form F = { ⊂ F i ⊂ · · · ⊂ F i r ⊂ R d } . The θ -flag variety F θ is the set of all θ -flags. Let a θ = (cid:8) ~a ∈ a | α k ( ~a ) = 0 if k / ∈ θ (cid:9) . There is a unique projection p θ : a → a θ OUNTING AND EQUIDISTRIBUTION 33 invariant by { w ∈ W : w ( a θ ) = a θ } where W is the Weyl group acting on a by coordinate permu-tations. Benoist and Quint [2, Section 8.6] describe a partial Iwasawa cocycle B θ : SL ( d, R ) × F θ → a θ such that p θ ◦ B factors through B θ .We say that A ∈ SL ( d, R ) is θ -proximal if α k ( ℓ ( A )) > k ∈ θ . In this case, A has awell-defined attracting θ -flag F θA , and B θ ( A, F θA ) = p θ ( ℓ ( A ))In particular,(9.2) ω k ( B θ ( A, F θA )) = ω k ( ℓ ( A ))for all k ∈ θ .Given a representation ρ : Γ → SL ( d, R ) of a geometrically finite Fuchsian group Γ and a ρ -equivariant map ξ ρ : Λ(Γ) → F θ we define its associated θ -Busemann cocycle β θρ : Γ × Λ(Γ) → a θ by letting β θρ ( γ, z ) = B θ (cid:0) ρ ( γ ) , ρ ( γ − )( ξ ρ ( z )) (cid:1) . Since p θ is linear, Lemma 9.5 immediately generalizes to give Lemma 9.6. If ρ : Γ → SL ( d, R ) is a representation of a geometrically finite Fuchsian group Γ and ξ : Λ(Γ) → F θ is a ρ -equivariant map, then β θρ satisfies the cocycle property β θρ ( αγ, z ) = β θρ ( α, z ) + β θρ ( γ, α − ( z )) for all α, γ ∈ Γ and z ∈ Λ(Γ) .Moreover, if ρ ( γ ) is θ -proximal and ξ ρ ( γ + ) is the attracting θ -flag of ρ ( γ ) , then β θρ ( γ, γ + ) = p θ ( ℓ ( ρ ( γ ))) . In particular, ω k ( β θρ ( γ, γ + )) = ω k ( ℓ ( ρ ( γ ))) if k ∈ θ . Roof functions for Anosov representations If θ ⊂ { , . . . , d − } is non-empty, we will say that ρ : Γ → SL ( d, R ) is cusped θ -Anosov if it iscusped P k -Anosov for all k ∈ θ . We say that θ is symmetric if k ∈ θ if and only if d − k ∈ θ . Itwill be natural to always assume that θ is symmetric, since ρ is cusped P k -Anosov if and only if itis cusped P d − k -Anosov. If ρ : Γ → SL ( d, R ) is a cusped θ -Anosov representation of a geometricallyfinite Fuchsian group, we define a vector valued roof function τ ρ : Σ + → a θ by setting τ ρ ( x ) = β θρ (cid:0) G ( x ) , ω ( x ) (cid:1) = B θ (cid:16) ρ ( G ( x )) , ρ ( G ( x )) − (cid:0) ξ ρ ( ω ( x )) (cid:1)(cid:17) . If φ is a linear functional on a θ we define the φ -roof function τ φρ = φ ◦ τ ρ . If ρ is cusped BorelAnosov, i.e. if θ = { , . . . , d − } , then a θ = a and B θ = B so we are in the simpler setting describedin the first part of Section 9.5.Recall that the Benoist limit cone of a representation ρ : Γ → SL ( d, R ) is given by B ( ρ ) = \ n ≥ [ || κ ( ρ ( γ )) ||≥ n R + κ ( ρ ( γ )) ⊂ a + . OUNTING AND EQUIDISTRIBUTION 34
Benoist [1] showed that if Γ is Zariski dense, then B ( ρ ) is convex and has non-empty interior. It isnatural to consider linear functionals which are positive on the Benoist limit cone B ( ρ ) + = n φ ∈ a ∗ | φ (cid:16) B ( ρ ) − { ~ } (cid:17) ⊂ (0 , ∞ ) o . Note that if φ ∈ B ( ρ ) + , then there is a constant c such that φ ( v ) > c k v k for all v ∈ B ( ρ ).We will in general consider roof functions associated to linear functionals in a ∗ θ ∩ B ( ρ ) + . Recallthat a ∗ θ is spanned by { ω k | k ∈ θ } . So if { , d − } ⊂ θ and ρ is cusped θ -Anosov (i.e. if ρ is cusped P -Anosov), then ω and the Hilbert length functional α H = ω + ω d − both lie in a ∗ θ ∩ B ( ρ ) + . If { , } ⊂ θ , then α = ω − ω ∈ a ∗ θ ∩ B ( ρ ) + , and, more generally, if { k − , k, k + 1 } ⊂ θ , then α k = − ω k +1 + 2 ω k − ω k − ∈ a ∗ θ ∩ B ( ρ ) + , if ρ is cusped θ -Anosov. Finally, if θ = { , . . . , d − } (i.e. ρ is cusped Borel Anosov), then∆ = (cid:8) a α + . . . + a d − α d − | a i ≥ ∀ i, d − X i =1 a i > (cid:9) ⊂ a ∗ θ ∩ B ( ρ ) + = B ( ρ ) + . Theorem D*:
Suppose that Γ is a torsion-free geometrically finite, but not convex cocompact,Fuchsian group, θ ⊂ { , . . . , d − } is non-empty and symmetric, and ρ : Γ → SL ( d, R ) is cusped θ -Anosov. If φ ∈ a ∗ θ ∩ B ( ρ ) + , then τ φρ : Σ + → R is a locally H¨older continuous function such that (1) If x = x · · · x n is a periodic element of Σ + , then S n τ φρ ( x ) = φ (cid:16) ℓ (cid:0) ρ ( G ( x ) · · · G ( x n )) (cid:1)(cid:17) . (2) τ φρ is eventually positive. (3) There exists C ρ > such that if j ∈ θ , then (cid:12)(cid:12)(cid:12) τ ω j ρ ( x ) − c j ( ρ, s ( x )) log r ( x ) (cid:12)(cid:12)(cid:12) ≤ C ρ (with the convention that c j ( ρ, γ ) = 0 if γ is not parabolic). (4) τ φρ has a strong entropy gap at infinity. Moreover, if φ = P k ∈ θ a k ω k , then d ( τ φρ ) = 1 c ( ρ, φ ) where c ( ρ, φ ) = inf n X k ∈ θ a k c k ( ρ, υ ) | υ ∈ Γ parabolic o . (5) If η ∈ Hom tp ( ρ ) is also P k -Anosov and φ ∈ B ( η ) + , then there exists C > so that | τ φρ ( x ) − τ φη ( x ) | ≤ C for all x ∈ Σ + . (6) τ φρ is non-arithmetic.Proof of Theorem D*. It follows immediately from Lemma 9.6 and Theorem 9.3 (1) that if x ∈ Σ + ,then(10.1) S n τ ρ ( x ) = n − X j =0 τ ρ ( σ j ( x )) = β θρ (cid:0) G ( x ) · · · G ( x m ) , ω ( x ) (cid:1) . In particular, if x = x · · · x n ∈ Σ + is periodic, then, by Lemma 9.6 and Theorem 9.3 (2),(10.2) ω k (cid:0) S n τ ρ ( x ) (cid:1) = ω k (cid:16) ℓ (cid:0) ρ ( G ( x ) · · · G ( x n )) (cid:1)(cid:17) , for all k ∈ θ , since ξ ρ ( ω ( x )) is the attracting θ -flag of ρ ( G ( x ) · · · G ( x n )). Thus, (1) holds since { ω k | k ∈ θ } is a basis for a ∗ θ and the map φ → τ φ is linear. OUNTING AND EQUIDISTRIBUTION 35 If φ ◦ τ ρ is not eventually positive, then there exist sequences { x n } in Σ + and { m n } in N so that m n → ∞ and φ (cid:0) S m n τ ρ ( x n ) (cid:1) < n . Let γ n = G (( x n ) ) · · · G (( x n ) m n ) and z n = ω ( x n ). Then φ (cid:0) β θρ ( γ n , z n ) (cid:1) < n ∈ N . We may assume that { z n } converges to z ∈ Λ(Γ). Theorem 9.3 (5) implies that there exists L so that d ( γ n ( b ) , −−→ b z n ) ≤ L for all n . After passing to another subsequence, we may assumethat { γ − n ( b ) } converges to some w ∈ Λ(Γ). We pass to another subsequence, so that { γ − n ( z n ) } converges to some x ∈ Λ(Γ). Notice that x = w , since −−−−−−−−−−−→ γ − n ( b ) γ − n ( z n ) converges to a bi-infinitegeodesic joining w to x which lies within L of the basepoint b .Since lim γ − n ( b ) = w and ρ has the P k -Cartan property for all k ∈ θ by Theorem 9.2(6),lim U k ( ρ ( γ − n )) = ξ kρ ( w ) . Since ξ d − kρ ( x ) and ξ d − kρ ( w ) are transverse, there exist N ∈ N and ǫ > n > N , then ∠ (cid:0) ξ kρ ( γ − n z n ) , U d − k ( ρ ( γ n ) − ) (cid:1) ≥ ǫ. Lemma 9.4 and the ρ -equivariance of the limit map ξ ρ then imply that there exists C so that | ω k ( β θρ ( γ n , ξ ρ ( z n ))) − ω k ( κ ( ρ ( γ n ))) | = | ω k ( B θ ( ρ ( γ n ) , ρ ( γ − n )( ξ ρ ( z n )))) − ω k ( κ ( ρ ( γ n ))) | ≤ C for all k ∈ θ and all n ≥ N . Since φ ∈ a ∗ θ this implies that there exists ˆ C > | φ ( β θρ ( γ n , ξ ρ ( z n ))) − φ ( κ ( ρ ( γ n ))) | ≤ ˆ C for all n ≥ N .By Theorem 9.1(1), φ ( κ ( ρ ( γ n ))) → ∞ , so we have achieved a contradiction. Therefore, τ φρ iseventually positive, so (2) holds.In order to establish (3), we first notice that, since || B θ ( A, F ) || ≤ || κ ( A ) || for all F ∈ F θ , | τ ω j ρ ( x ) | ≤ C x = j || κ ( ρ ( G ( x ))) || for all x ∈ Σ + and j ∈ θ . Since our alphabet is infinite and C x → ∞ as r ( x ) → ∞ , there is morework to be done.If x ∈ Σ + and r ( x ) ≥
2, then G ( x ) = υ n g a for some υ ∈ P and g a ∈ R , where n = r ( x ) − τ ρ ( x ) = β θρ (cid:0) υ n g a , ω ( x ) (cid:1) = B θ (cid:0) ρ ( υ n g a ) , ρ ( υ n g a ) − ( ξ ρ ( ω ( x ))) (cid:1) = B θ (cid:0) ρ ( υ n ) , ρ ( υ − n )( ξ ρ ( ω ( x ))) (cid:1) + B θ (cid:0) ρ ( g a ) , ρ ( υ n g a ) − ( ξ ρ ( ω ( x ))) (cid:1) . Notice that (cid:12)(cid:12)(cid:12) ω j (cid:16) B θ (cid:0) ρ ( g a ) , ρ ( υ n g a ) − ( ξ ρ ( ω ( x ))) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) ≤ R = max (cid:8) d k κ ( ρ ( g a )) k (cid:12)(cid:12) g a ∈ R (cid:9) for all j ∈ θ .Let p be the fixed point of υ in Λ(Γ). Notice that, by construction, there exists ˆ a ∈ A so that G (ˆ a ) = υg a . Then X = ω ([ˆ a ]) is a compact subset of Λ(Γ) − { p } . Therefore, if G ( x ) = υ n g a , ω ( x ) ∈ υ n − ( X ), so υ − n ( ω ( x )) ∈ υ − ( X ). It follows that there exists ǫ = ǫ ( υ ) > G ( x ) = υ n g a and n ∈ N , then ∠ (cid:0) ρ ( υ − n )( ξ jρ ( ω ( x ))) , ξ d − jρ ( p ) (cid:1) ≥ ǫ for all j ∈ θ . Lemma 9.4 then implies that there exists D = D ( υ, g a ) > (cid:12)(cid:12)(cid:12) ω j (cid:0) B θ (cid:0) ρ ( υ n ) , ρ ( υ − n )( ξ ρ ( ω ( x ))) (cid:1) − ω j ( κ ( ρ ( υ n ))) (cid:12)(cid:12)(cid:12) ≤ D. for all n ∈ N and j ∈ θ . Theorem 9.1 implies that there exists C = C ( υ, g a ) > (cid:12)(cid:12) ω j ( κ ( ρ ( υ n ))) − c j ( ρ, υ ) log n (cid:12)(cid:12) < C OUNTING AND EQUIDISTRIBUTION 36 for all n ∈ N . By combining, we see that (cid:12)(cid:12)(cid:12) ω j (cid:16) B θ (cid:0) ρ ( υ n ) , ρ ( υ − n )( ξ ρ ( ω ( x ))) (cid:1)(cid:17) − c j ( ρ, υ ) log n (cid:12)(cid:12)(cid:12) ≤ C + D and hence that (cid:12)(cid:12)(cid:12) τ ω j ρ ( x ) − c j ( ρ, υ ) log (cid:0) r ( x ) − (cid:1)(cid:12)(cid:12)(cid:12) ≤ C + D + R for all n ∈ N and j ∈ θ . Since there are only finitely many υ in P , and only finitely many elementsof A so that r ( a ) ≤ τ φρ is locally H¨older continuous. Since ω : Σ + → Λ(Γ) is locally H¨oldercontinuous, there exist
Z > ζ > x j = y j for all j ≤ n , then d ( ω ( x ) , ω ( y )) ≤ Ze − ζn . Since ξ ρ : Λ(Γ) → F d is H¨older, there exist D > ι >
0, so that if z, w ∈ Λ(Γ), then d ( ξ ρ ( z ) , ξ ρ ( w )) ≤ Dd ( z, w ) ι Therefore, ξ ρ ◦ ω is locally H¨older continuous, i.e. there exists C and β > d ( ξ ρ ( ω ( x )) , ξ ρ ( ω ( y ))) ≤ Ce − βn if x j = y j for all j ≤ n .If a ∈ A , let D a = sup n || D F B θ ( ρ ( G ( a )) , · ) || (cid:12)(cid:12)(cid:12) F ∈ F θ o where D F B θ ( ρ ( G ( a )) , · ) is the derivative at F of B θ ( ρ ( G ( a )) , · ) : F θ → a θ . It follows that if x j = y j for all j ≤ n and x = y = a , then | τ φρ ( x ) − τ φρ ( y ) | ≤ || φ || D a Ce − βn Recall that if x ∈ Σ + and G ( x ) = υ m g a , then τ ρ ( x ) = B θ (cid:0) ρ ( υ m ) , ρ ( υ − m )( ξ ρ ( ω ( x ))) (cid:1) + B θ (cid:0) ρ ( g a ) , ρ ( υ m g a ) − ( ξ ρ ( ω ( x ))) (cid:1) and that υ − m ( ω ( x )) lies in a compact subset υ − ( X ) of Λ(Γ) − { p } (where p is the fixed point of υ ).There exists c > x, y ∈ υ − ( X ) and r ∈ N , then d ( υ r ( x ) , υ r ( y )) ≤ cr d ( x, y ) . Notice that, by the cocycle property for B θ , B θ (cid:0) ρ ( υ m ) , F (cid:1) = m X j =1 B θ ( ρ ( υ ) , υ j − ( F )) . Thus, if ˆ D = ˆ D ( υ ) = sup n || D F B θ ( ρ ( υ ) , · ) || (cid:12)(cid:12)(cid:12) F ∈ F θ o then || B θ (cid:0) ρ ( υ m ) , x (cid:1) − B θ (cid:0) ρ ( υ m ) , y (cid:1) || ≤ m X s =1 ˆ D cs d ( x, y )if x, y ∈ υ − ( X ). Notice that there exists T = T ( υ ) > T d ( x, y ). Therefore, if x j = y j for all j = 1 , . . . , n and G ( x ) = υ s g a where s ≥
1, then | ( φ ◦ τ ρ )( x ) − ( φ ◦ τ ρ )( y ) | ≤ ( T + R ) C || φ || e − βn where R = sup n || D F B θ ( ρ ( g a ) , · ) || (cid:12)(cid:12)(cid:12) F ∈ F d , g a ∈ R o . OUNTING AND EQUIDISTRIBUTION 37
Since there are only finitely many υ in P and only finitely many elements of A so that r ( a ) ≤ τ φρ is locally H¨older continuous.If φ = P k ∈ θ a k ω k and υ ∈ P , let c ( ρ, φ, υ ) = X k ∈ θ a k c k ( ρ, υ ) and c ( ρ, φ ) = inf { c ( ρ, φ, υ ) | υ ∈ P} . Notice that c ( ρ, φ ) must be positive, since φ ∈ B ( ρ ) + . Property (3) then implies that (cid:12)(cid:12) τ φρ ( x ) − c ( ρ, φ, s ( x )) log( r ( x )) (cid:12)(cid:12) ≤ C ρ || φ || for all x ∈ Σ + . Therefore, ∞ X n =1 e − sC ρ || φ || n sc ( ρ,φ ) = ∞ X n =1 e − s (cid:0) c ( ρ,φ ) log n + C ρ || φ || (cid:1) ≤ Z ( τ φρ , s )and Z ( τ φρ , s ) ≤ ∞ X n =1 Q P ) e − s (cid:0) c ( ρ,φ ) log n − C ρ || φ || (cid:1) ≤ ∞ X n =1 Q P ) e sC ρ || φ || n sc ( ρ,φ ) if s >
0. (Recall that if n ∈ N , then 1 ≤ { a ∈ A | r ( a ) = n } ≤ Q .) Therefore, Z ( τ φρ , s ) convergesif and only if s > c ( ρ,φ ) , which establishes (4).If η ∈ Hom tp ( ρ ) is cusped θ -Anosov and φ ∈ B ( η + ), then c j ( ρ, υ ) = c j ( η, υ ) for all j ∈ θ and υ ∈ P . Property (5) then follows from applying (3) to both τ ρ and τ η and the fact that both τ φρ and τ φη are locally H¨older continuous.If the Zariski closure of ρ (Γ) is not reductive, then Gu´eritaud-Guichard-Kassel-Wienhard [21,Section 2.5.4] exhibit a representation ρ ss : Γ → SL ( d , R ) so that the Zariski closure G of ρ ss (Γ)is reductive and ℓ ( ρ ( γ )) = ℓ ( ρ ss ( γ )) for all γ ∈ Γ. A result of Benoist-Quint [2, Proposition 9.8]implies that the subgroup h of the Cartan algebra a g of G generated by λ G ( ρ (Γ)) is dense in a g (where λ G : G → a g is the Jordan projection of G ). Up to conjugation, we may assume that a g is a sub-algebra of a (since a g is an abelian algebra and thus is contained in a translate of a ,which is a maximal abelian sub-algebra of sl ( d, R )). Therefore, the subgroup of R generated by { φ ◦ τ ρ ( x ) | x ∈ Fix n } , which is just φ ( h ), is dense in R . Thus, we have established (6). (cid:3) Applications
Anosov representations of geometrically finite Fuchsian groups.
Given Theorem D*,we can apply our main results to the roof functions of Anosov representations.The following counting result is a strict generalization of Corollary 1.3. It follows immediatelyfrom Theorems D* and A.
Corollary 11.1.
Suppose that Γ is a torsion-free, geometrically finite, but not convex cocompact,Fuchsian group, θ ⊂ { , . . . , d − } is non-empty and symmetric, and ρ : Γ → SL ( d, R ) is cusped θ -Anosov. If φ ∈ a ∗ θ ∩ B ( ρ ) + , then there exists a unique δ φ ( ρ ) > c ( ρ,φ ) so that P ( − δ φ ( ρ ) τ φρ ) = 0 and lim t →∞ M φ ( t ) tδ φ ( ρ ) e tδ φ ( ρ ) = 1 where M φ ( t ) = n [ γ ] ∈ [Γ] (cid:12)(cid:12) < φ ( ℓ ( ρ ( γ ))) ≤ t o . Similarly, one may combine Theorems C and D* to obtain a generalization of Corollary 1.4.
OUNTING AND EQUIDISTRIBUTION 38
Corollary 11.2.
Suppose that Γ is a torsion-free, geometrically finite, but not convex cocompact.Fuchsian group, θ ⊂ { , . . . , d − } is non-empty and symmetric, and ρ : Γ → SL ( d, R ) is cusped θ -Anosov. If η ∈ Hom tp ( ρ ) is also cusped θ -Anosov, φ ∈ a ∗ θ ∩ B ( ρ ) + ∩ B ( η ) + , and C φ ( ρ, η ) = (cid:8) ( a, b ) ∈ D ( ρ, η ) | P ( − aτ φρ − bτ φη ) = 0 (cid:9) where D ( ρ, η ) = (cid:8) ( a, b ) ∈ R | a + b > c ( ρ, φ ) (cid:9) , then (1) C φ ( ρ, η ) is an analytic curve, (2) ( δ φ ( ρ ) , and (0 , δ φ ( η )) lie on C φ ( ρ, η ) , (3) C ( ρ, η ) is strictly convex, unless ℓ φ ( ρ ( γ )) = δ φ ( η ) δ φ ( ρ ) ℓ φ ( η ( γ )) for all γ ∈ Γ , (4) and the tangent line to C φ ( ρ, η ) at ( δ φ ( ρ ) , has slope s φ ( ρ, η ) = − R τ φη dm − δ φ ( ρ ) τ φρ R τ φρ dm − δ φ ( ρ ) τ φρ . In the setting of the previous corollary, we may define the pressure intersection I φ ( ρ, η ) = − s φ ( ρ, η )and the renormalized pressure intersection J φ ( ρ, η ) = δ φ ( η ) δ φ ( ρ ) I φ ( ρ, η ) . We obtain the following intersection rigidity result which will be used crucially in the constructionof pressure metrics. The proof follows at once from statements (3) and (4) in Corollary 11.2.
Corollary 11.3.
Suppose that Γ is a torsion-free, geometrically finite, but not convex cocompact,Fuchsian group, θ ⊂ { , . . . , d − } is non-empty and symmetric, and ρ : Γ → SL ( d, R ) is cusped θ -Anosov. If η ∈ Hom tp ( ρ ) is also cusped θ -Anosov and φ ∈ a ∗ θ ∩ B ( ρ ) + ∩ B ( η ) + , then J φ ( ρ, η ) ≥ with equality if and only if ℓ φ ( ρ ( γ )) = δ φ ( η ) δ φ ( ρ ) ℓ φ ( η ( γ )) for all γ ∈ Γ . Finally, we derive our equidistribution result, which generalizes Corollary 1.6. It follows imme-diately from Theorems B and D*.
Corollary 11.4.
Suppose that Γ is a torsion-free, geometrically finite, but not convex cocompact,Fuchsian group, θ ⊂ { , . . . , d − } is non-empty and symmetric, and ρ : Γ → SL ( d, R ) is cusped θ -Anosov. If η ∈ Hom tp ( ρ ) is also cusped θ -Anosov and φ ∈ a ∗ θ ∩ B ( ρ ) + ∩ B ( η ) + , then I φ ( ρ, η ) = lim T →∞ R φT ( ρ )) X [ γ ] ∈ R φT ( ρ ) ℓ φ ( η ( γ )) ℓ φ ( ρ ( γ )) where R T ( ρ ) = { [ γ ] ∈ Γ | < ℓ φ ( ρ ( γ )) ≤ T } . OUNTING AND EQUIDISTRIBUTION 39
Traditional Anosov representations.
Andres Sambarino [52, 53, 54] established ana-logues of our counting and equidistribution results in the setting of traditional “uncusped” Anosovrepresentations. In this section, we will sketch how to establish (mild generalizations of) his re-sults in our framework. We start by recalling a characterization of Anosov representations of wordhyperbolic groups established by Kapovich-Leeb-Porti [29] and Bochi-Potrie-Sambarino [4].If Γ is a word hyperbolic group, then a representation ρ : Γ → SL ( d, R ) is P k -Anosov if thereexist A, a > σ k ( ρ ( γ )) σ k +1 ( ρ ( γ )) ≥ Ae a | γ | for all γ ∈ Γ, where | γ | is the word length of γ with respect to some fixed generating set on Γ. Inthis case, it is known (see [10] or [15]) that there is a finite Markov shift (Σ +Γ , σ ) for the geodesicflow U Γ of Γ and a surjective map G : [ n ∈ N Fix n → [Γ] . Moreover, if θ ⊂ { , . . . , d − } is non-empty and symmetric, ρ is θ -Anosov, and φ ∈ a θ ∩ B ( ρ ) + ,then there exists a H¨older continuous function τ φρ : Σ +Γ → R so that if x ∈ Fix n ⊂ Σ +Γ , then S n τ φρ ( x ) = φ ( ℓ ( ρ ( G ( x )))) . Lalley [33, Theorems 5 and 7] established analogues of our counting and equidistribution resultsfor finite Markov shifts. Moreover, our proofs generalize his techniques so they go through in thesetting of finite Markov shifts without any assumptions on entropy gap.
Corollary 11.5.
Suppose that Γ is a word hyperbolic group, θ ⊂ { , . . . , d − } is non-empty andsymmetric, and ρ : Γ → SL ( d, R ) is θ -Anosov. If φ ∈ a ∗ θ ∩ B ( ρ ) + , then there exists a unique δ φ ( ρ ) > so that P ( − δ φ ( ρ ) τ φρ ) = 0 and lim t →∞ M φ ( t ) tδ φ ( ρ ) e tδ φ ( ρ ) = 1 where M φ ( t ) = n [ γ ∈ [Γ] (cid:12)(cid:12) φ ( ℓ ( ρ ( γ ))) ≤ t o . Proof.
Our proof of property (6) in Theorem D* gives immediately that τ φρ is non-arithmetic, whichis the only assumption needed to apply our Theorem A or Theorem 7 in [33] in the setting of afinite Markov shifts. (cid:3) We also obtain a Manhattan Curve theorem, which does not seem to have appeared in printbefore in this generality, but was certainly well-known to experts. In particular, Sambarino [53,Proposition 4.7] describes a closely related phenomenon for Borel Anosov representations.
Corollary 11.6.
Suppose that Γ is a word hyperbolic group, θ ⊂ { , . . . , d − } is non-empty andsymmetric, and that ρ : Γ → SL ( d, R ) and η : Γ → SL ( d, R ) are θ -Anosov. If φ ∈ a ∗ θ ∩ B ( ρ ) + ∩ B ( η ) + and C φ ( ρ, η ) = (cid:8) ( a, b ) ∈ R | a + b > P ( − aτ φρ − bτ φη ) = 0 (cid:9) , then (1) C φ ( ρ, η ) is an analytic curve, (2) ( δ φ ( ρ ) , and (0 , δ φ ( η )) lie on C φ ( ρ, η ) , (3) and C φ ( ρ, η ) is strictly convex, unless ℓ φ ( ρ ( γ )) = δ φ ( η ) δ φ ( ρ ) ℓ φ ( η ( γ )) for all γ ∈ Γ . OUNTING AND EQUIDISTRIBUTION 40
Moreover, the tangent line to C φ ( ρ, η ) at ( δ φ ( ρ ) , has slope − I φ ( ρ, η ) = − R τ φη dm − δ φ ( ρ ) τ φρ R τ φρ dm − δ φ ( ρ ) τ φρ The analogues of Corollaries 1.5 and 1.6 appear in [10, Section 8] as consequences of classicalThermodynamical results of Bowen, Pollicott and Ruelle [5, 6, 44, 50].
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