EXPONENTIAL ERROR TERMS FOR GROWTH FUNCTIONS ON NEGATIVELY CURVED SURFACES By MARK POLLICOTT and RICHARD SHARP

In this paper we consider two counting problems associated with compact negatively curved surfaces and improve classical asymptotic estimates due to Margulis. In the first, we show that the number of closed geodesics of length at most T has an exponential error term. In the second we show that the number of geodesic arcs (between two fixed points x and y) of length at most T has an exponential error term. The proof is based on a detailed study of the zeta function and Poincaré series and benefits from recent work of Dolgopiat. 0. Introduction. Let V be a compact surface with negative curvature and let Ṽ be its universal cover (with the lifted metric d). The fundamental group 1(V) can be realized as a group of isometries Γ acting on Ṽ with Ṽ=Γ = V . Given a point x 2 Ṽ , we can introduce the orbital counting function N(T) defined by N(T) = Cardfg 2 Γ: d(x, gx) Tg. The compact surface V has a countable infinity of closed geodesics (of length l( )) and we can also introduce the closed geodesic counting function (T) = Cardf : l( ) Tg. For the special case of surfaces of constant negative curvature there are results due to Huber [7, 8] giving both an asymptotic formula and error terms for N(T) and (T), as T tends to infinity. In the more general case of variable negative curvature Margulis again established an asymptotic formula for N(T) and (T) but without error terms [10]. While in the constant curvature case one can make use of harmonic analysis and the Selberg Trace Formula, in the more general setting it is necessary to resort to an approach based on the dynamics of the geodesic flow. In this paper, we shall improve Margulis’s original estimates to include the appropriate error terms. We can associate to V the geodesic flow t: SV ! SV , where SV is the unit tangent bundle of V . We shall let h > 0 denote the topological entropy of . Manuscript received September 23, 1997. American Journal of Mathematics 120 (1998), 1019–1042. 1019 1020 MARK POLLICOTT AND RICHARD SHARP Then h is also equal to the exponential growth rate of the area of balls in Ṽ , i.e., h = lim T! +1 1 T log Areafy 2 Ṽ: d(x, y) Tg. Notation. We write f (T) g(T) if f (T)=g(T) ! 1 as T ! +1. We write f (T) = g(T) + O(h(T)) if there exists C > 0 such that j f (T) g(T)j Ch(T). The results of Margulis [10] show that N(T) and (T) have the following first order asymptotic expansions, as T ! +1: N(T) Ce and (T) li(e), where C > 0 is a constant and where li(y) is the logarithmic integral li(y) = Z y 2 1 log u du y log y , as y ! +1. In this paper we show that the error terms take the form given in the following two theorems. THEOREM 1. There exists 0 < c < h such that (T) = li(ehT) + O ecT . THEOREM 2. There exists 0 < c < h and C > 0 such that N(T) = CehT +O(ecT). We now give a brief outline of the contents of the paper. In Section 1, we discuss symbolic coding of the geodesic flow from two viewpoints. The first is a geometric coding best suited to studying N(T) and the second is a dynamical coding better suited to studying (T). In Section 2, we give a brief account of Dolgopiat’s recent work on transfer operators. In Section 3 we use estimates on transfer operators to obtain estimates for a certain zeta function. In Section 4 we complete the proof of Theorem 1. In Section 5 we introduce the Poincaré series associated to orbit counting and a modified transfer operator. In Section 6 prove the necessary results on the associated Poincaré series to complete the proof of Theorem 2. 1. Symbolic coding for the geodesic flow. In this section we shall describe two methods for coding the geodesic flow and the discrete group Γ. The first approach is the geometric coding obtained by Bowen and Series of Γ and its limit set. The second is the Bowen-Ratner coding for the geodesic flow. 1.1. The geometric coding. Usually the Bowen-Series coding is only applied to geodesic flows on surfaces of constant negative curvature. However, it is an easy observation that it can be adapted to study geodesic flows on surfaces of variable curvature. This is easily seen because every geodesic flow with respect GROWTH FUNCTIONS ON NEGATIVELY CURVED SURFACES 1021 to a metric of variable negative curvature is flow equivalent (i.e. conjugate up to a time change) to one of constant negative curvature [6]. Suppose that V has genus g 2, then Γ = 1(V) has the one relator presentation * a1, : : : , ag, b1, : : : , bg: g Y i=1 [ai, bi] = 1 + . We can associate to this presentation a finite directed graph G with edge set E and vertex set V together with: (1) a distinguished vertex with no edges ending at ; (2) a labeling of the edges : E ! Γ0 = fa1, : : : , ag, b1, : : : , bgg, such that there is a bijection between (i) the set of finite paths starting at ; and (ii) elements of Γ, given by multiplying the labels on each edge. Furthermore, G has the important additional property that the map from closed loops to conjugacy classes in Γ obtained by considering the product of the labels around the loop is a bijection [20, 21]. For technical reasons it will prove useful to consider an augmented graph G0. This is obtained by adding an extra vertex 0 and edges (i, 0), for all i 2 V f g. We extend the map to the new edges by (i, 0) = e, the identity in Γ. Let A denote the incidence matrix of G0. Let B denote the submatrix of A obtained by deleting the rows and columns indexed by and 0. We can associate one-sided subshifts of finite type : X+ A ! X + A and : X + B ! X+ B where X A = ( x = (xn) 1 n=0 2 1 Y n=0 (V [ f0g): A(xn, xn+1) = 1,8n 0 ) ; X B = ( x = (xn) 1 n=0 2 1 Y n=0 V f g : B(xn, xn+1) = 1,8n 0 ) and (xn)n=0 = (xn+1) 1 n=0. We can define metrics on X + A and X + B by d((xn) 1 n=0, (yn) 1 n=0) = 1 X