Vadim A. KAIMANOVICH 0. Introduction - CiteSeerX

ERGODICITY OF HARMONIC INVARIANT MEASURES FOR THE GEODESIC FLOW ON HYPERBOLIC SPACES Vadim A. KAIMANOVICH 0. Introduction

The notions of ergodicity (absence of non-trivial invariant sets) and conservativity (absence of non-trivial wandering sets) are basic for the theory of measure preserving transformations. Ergodicity implies conservativity, but the converse is not true in general. Nonetheless, transformations from some classes always happen to be either ergodic (hence, conservative), or completely dissipative (i.e., their ergodic components are just orbits in the state space). The rst statement of this type was proved by Hopf [Ho] for the geodesic ow on Riemannian surfaces with constant negative curvature (whence the name \the Hopf dichotomy"). It was generalized by Sullivan to the geodesic ow on higher dimensional Riemannian manifolds with constant negative curvature [Su2] (both Hopf and Sullivan considered the Liouville invariant measure of the geodesic ow). It turned out that ergodicity of the geodesic ow is equivalent to recurrence of the Brownian motion on the manifold (in this form it is called the Hopf{Tsuji{Sullivan theorem). Later Sullivan generalized this result to invariant measures arising from a conformal density on the sphere at in nity [Su3] (see also a survey in [N] and a recent paper [Y]). Of course, the Liouville measure is not the only invariant measure of the geodesic

ow. Another natural invariant measure of the geodesic ow on a negatively curved Riemannian manifold with pinched sectional curvatures is the harmonic invariant measure connected with the Brownian motion (see [K1]). The harmonic invariant measure coincides with the Liouville measure in the constant curvature case, or, more generally, for rank 1 locally symmetric manifolds. Under some additional assumptions, the harmonic invariant measure can be associated with any Markov operator on a Gromov hyperbolic space. The aim of this paper is to prove an analogue of the Hopf dichotomy for the harmonic measure in this generality: either the quotient Markov operator is recurrent and the geodesic ow is ergodic with respect to the harmonic invariant measure, or the quotient operator is transient and the geodesic ow is completely dissipative. Although the general scheme of our proof is the same as in Sullivan's papers [Su2], [Su3], an auxiliary intermediate condition being that the harmonic measure is concentrated on the radial limit set, the presentation and the details are dierent. Supported by a British SERC Advanced Fellowship. A part of this work was done during my visit to the University of Strasbourg supported by CNRS. 1991 Mathematics Subject Classi cation. Primary 28D10, 57S25, 58F11, 58F17, 60J50; Secondary 05C05, 31C35. Typeset by AMS-TEX 1

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VADIM A. KAIMANOVICH

We also show that for the geodesic ow on a quotient of a Gromov hyperbolic space the Hopf dichotomy holds true for any invariant measure corresponding to a quasiproduct geodesic current. We are dealing with general hyperbolic spaces, nonetheless our proofs are (hopefully) simpler than even in the constant curvature case, which is achieved by using general methods of ergodic theory and theory of Markov operators. We tried to keep the exposition as self-contained as possible, so that the reader who is interested only in the Hopf{Tsuji{Sullivan theorem is invited to read this paper having in mind the hyperbolic n-space H n (instead of a general Gromov hyperbolic space) with the Liouville invariant measure of the geodesic ow (which coincides in this case with the harmonic invariant measure of the Brownian motion). The paper has the following structure. Section 1 is devoted to general ergodic properties of Markov operators with a stationary measure. Most of the contents of x1.1 and x1.2 should be known to specialists (e.g., see [Kr2], [Fo], [Re], [CFS]). Nonetheless, it would be dicult to give exact references for some of the results we use. For this reason, and also in order to make exposition easier to follow and to reveal the \mechanics" of our approach which would have remained hidden otherwise, we give short proofs of some well known results. In x1.1 we discuss the notions of conservativity and dissipativity of general group actions and introduce the Hopf decomposition into conservative and completely dissipative parts. Theorem 1.1 shows equivalence of ergodicity of conservative endomorphisms and of their natural extensions. x1.2 is devoted to general ergodic properties of Markov operators . We introduce the reversed operator of a Markov operator with a stationary measure, and discuss the notions of irreducibility and recurrence. Further we de ne the Poisson boundary of a Markov operator as the space of ergodic components of the shift in the (unilateral) path space of a Markov operator (De nition 1.6). The Poisson boundary is a measure space and describes the (stochastically signi cant) behaviour of sample paths at in nity [K3]. Conditioning sample paths of the Markov chain both at ?1 and at +1 by xing points from the Poisson boundaries of the reversed and of the original operators gives a bilateral conditional decomposition of a Markov operator with stationary measure (Proposition 1.6). Combination of the Hopf dichotomy for Markov operators (equivalence of recurrence and conservativity) and of Theorem 1.1 gives the following dichotomy for irreducible Markov operators with a stationary measure (Theorem 1.2): the shift in their bilateral path space is either ergodic (in the recurrent case), or completely dissipative (in the transient case). In x1.3 we introduce the notion of a covering Markov operator which is a measure theoretical generalization of Markov operators corresponding to diusion processes on covering Riemannian manifolds (De nition 1.7, see also [K4]). The state space of a covering Markov operator is a covering measure space, i.e., a measure space with a measure preserving completely dissipative action of a countable group, and the operator itself is invariant with respect to the action of this group. The main result of x1.3 are Theorems 1.3 and 1.4 which show that for a corecurrent covering operator with a stationary measure the action of the deck group on the product of the Poisson boundaries of the original operator and its reversal is ergodic. In particular, if a corecurrent covering

ERGODICITY OF HARMONIC INVARIANT MEASURES

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operator is reversible, then the deck group action on the square of its Poisson boundary is ergodic. We give two proofs of this result. One is based on Theorem 1.2, whereas the other one is purely Markov and does not use any ergodic theory at all. In both these proofs we use extensions of the original covering Markov operator. The paths of one extension have the form fxn?1 ; xng, and the paths of the other one | the form fbnd ? (x); xng, where bnd ? (x) is the point of the Poisson boundary of the reversed operator corresponding to the path x = fxn g (the \limit point at ?1" of the path x). Both extensions are also covering Markov operators, and they are corecurrent if the original operator is. Section 2 is devoted to invariant measures of the geodesic ow on quotients of hyperbolic spaces. In x2.1 we give a de nition of Gromov hyperbolic spaces and introduce two conditions on these spaces: uniqueness of the geodesic joining any two boundary points (condition (U)), and convergence of any two asymptotic geodesic rays (condition (C)). These conditions are, in particular, satis ed for real trees and for Cartan{Hadamard manifolds with pinched sectional curvature. Any geodesic in a hyperbolic space X has two endpoints on the hyperbolic boundary @X , and, conversely, under condition (U) for any two distinct boundary points there is a unique geodesic joining these points. This leads to a convex isomorphism between the cone of Radon invariant measures of the geodesic ow on X and the cone of Radon measures on the \square of in nity" @ 2X = @X @X n diag (Theorem 2.1). Thus, for a covering hyperbolic space Xe with a deck group G there is a convex isomorphism between the cone of Radon invariant measures of the quotient geodesic ow and the cone of corresponding geodesic currents (i.e., G-invaraint Radon measures on @ 2Xe ). This correspondence preserves conservativity (resp., complete dissipativity) of the corresponding invariant measures (Theorem 2.2). In x2.3 we connect conservativity of the geodesic ow on a quotient hyperbolic space with the radial limit set r . Since conservativity of the geodesic ow is equivalent to conservativity of its time reversal (the idea to use this fact is due to Coornaert and Papadopoulos [CP]), the conservative part of any geodesic current is concentrated on

r r , and its dissipative part is concentrated on r r , where r = @ Xe n r (Theorem 2.3). If is a quasi-product geodesic current, i.e., ? + for two probability measures ? ; + on @ Xe , then it immediately gives the following dichotomy: either ? ( r ) = + ( r ) = 1, and the geodesic current and the corresponding invariant measure of the geodesic ow are conservative, or ? ( r ) = + ( r ) = 0, and the geodesic current and the corresponding invariant measure of the geodesic ow are completely dissipative (Theorem 2.4). Further, if the space Xe satis es both axioms (U) and (C), then for any quasi-product geodesic current its conservativity implies ergodicity (Theorem 2.5). The idea of using asymptotic convergence of geodesics to prove ergodicity of the geodesic ow goes back to Hopf [Ho], [Su2]. However, the usual approach is based on the ratio ergodic theorem for conservative ows which, in particular, causes some trouble with choosing an integrable reference function [Su3], [CP]. We use a dierent approach based on deducing ergodicity of the geodesic ow from ergodicity of its induced ows on nite measure subsets. So, one can use the usual ( nite measure) Birkho ergodic theorem instead of the ratio ergodic theorem.

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The class of quasi-product geodesic currents (or, of corresponding invariant measures of the geodesic ow) is suciently wide. In particular, for negatively curved Riemannian manifolds it contains three \most common" invariant measures: the Liouville measure, the harmonic measure, and the maximal entropy (or, the Bowen{Margulis) measure. The latter measure can be de ned for an arbitrary hyperbolic space using the notion of conformal density . Theorems 2.4 and 2.5 are applicable to geodesics currents arising from conformal densities on any covering hyperbolic space satisfying axioms (U) and (C) (Theorem 2.7). Another application is to convex cocompact groups , or, more generally, to word hyperbolic groups . These groups as hyperbolic spaces do not in general satisfy conditions (U) and (C), so that the geodesic ow on such groups can be de ned up to a \quasi cation" only [Gr]. However, the dichotomy of Theorems 2.4 and 2.5 for the corresponding quasi-product geodesic currents still holds true (Theorem 2.8). In x2.5 we discuss interrelations between the notions of hyperbolicity and visibility for a Cartan{Hadamard manifold X and prove that hyperbolicity is equivalent to uniform visibility (Theorem 2.9), and that if the curvature is uniformly bounded then hyperbolic and visibility topologies on @ 2 X coincide (Theorem 2.10). Thus, in the latter case Theorem 2.1 gives a convex correspondence between the Radon invariant measures of the geodesic ow on SX and the Radon (with respect to the visibility uniform structure) measures on @ 2X . Section 3 is devoted to the harmonic invariant measure of the geodesic ow on hyperbolic spaces. Ledrappier [Le] proposed an ergodic construction of a harmonic invariant measure for the geodesic ow on compact negatively curved manifolds (corresponding to the Brownian motion on its universal covering space), and proved its ergodicity (in fact, he showed that the harmonic invariant measure is a Gibbs measure in this case). Later another, more geometric construction (but also essentially using the compactness assumption) was given by Hamenstadt [Ha]. In [K1] a harmonic geodesic current was constructed for an arbitrary Cartan{Hadamard manifold with pinched sectional curvatures. The main idea of [K1] can be applied to a general Markov operator P with a stationary measure on a hyperbolic space X . It consists in considering the space of bilateral sample paths of the corresponding Markov chain. If a.e. path converges to the hyperbolic boundary both at ?1 and at +1, then replacing it with the geodesic joining the two limit points immediately gives an invariant measure of the geodesic ow. The main problem here is to show that this invariant measure is a Radon measure (Theorem 3.2). In order to do it one needs several conditions on the operator P , the principal one being almost multiplicativity of its Green function along all geodesics (condition (M)). This condition can be easily veri ed for rank 1 symmetric spaces (there the Green function is a function of distance between points, and is well-known to be asymptotically exponential). Conditions, under which (M) holds in the general situation were found by Ancona [An1], [An2]. In particular, (M) holds for the Markov operator of the Brownian motion on an arbitrary Cartan{Hadamard manifold with pinched sectional curvatures. Condition (M) implies that the Martin boundary of the operator P coincides with the hyperbolic boundary of the space X (Theorem 3.1). Thus, almost all paths of the corresponding Markov chain converge in the hyperbolic topology, and the Poisson boundary of the operator P coincides with the hyperbolic boundary considered as a measure space with the corresponding harmonic measure class. This result is also due to


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Ancona, but our exposition is slightly dierent, because we make emphasis on condition (M). So, our proof applies directly to situations where (M) can be obtained without making recourse to general methods of Ancona (e.g., to rank 1 symmetric spaces). Condition (M) also immediately implies an upper estimate for the harmonic measure of shadows on @X in terms of the Green function (Lemma 3.2). This is an analogue of the Hall Lemma well-known in analysis. The main result of the paper is Theorem 3.3 which gives a dichotomy for the harmonic invariant measure of the geodesic ow on a covering hyperbolic space. Either, the quotient Markov operator is recurrent, the harmonic measure class is supported by the radial limit set, and the geodesic ow is ergodic with respect to the harmonic invariant measure. Or, the quotient Markov operator is transient, the harmonic measure class is supported by the complement of the radial limit set, and the geodesic ow is completely dissipative with respect to the harmonic invariant measure. Indeed, if the quotient Markov operator is recurrent, then by Theorem 1.4 the corresponding geodesic current is ergodic, and the geodesic ow is also ergodic and conservative. Conversely, if the quotient Markov operator is transient, then Lemma 3.2 implies that the harmonic measure is concentrated on the complement of the radial limit set, so that the geodesic

ow is completely dissipative. Note that our proof of Theorem 3.3 does not use the general Theorem 2.5 and can be obtained without any use of the Birkho ergodic theorem and of almost any ergodic considerations at all (because the proof of Theorem 1.4 also can be obtained by purely Markov means). Thus, for the geodesic ow on Riemannian manifolds of constant negative curvature (or, more generally, on rank 1 locally symmetric spaces) with the Liouville invariant measure it gives a proof of the Hopf{Tsuji{Sullivan theorem without any use of the Birkho ergodic theorem (contrary to a common opinion, e.g., see [Ah]). The only property of rank one symmetric spaces which is required for this proof is an estimate for the harmonic measure of shadows (Lemma 3.2) which can be veri ed straightforwardly. This paper was stumilated by a question asked by Terry Lyons. I am also grateful to him for numerous interesting discussions. 1. Ergodic properties of Markov operators with a stationary measure

1.1. Conservativity and dissipativity of group actions

1.1.1. General actions De nition 1.1. Let ( ; m) be a measure space with a measure type preserving action of a countable group G (i.e., the measure m is quasi-invariant with respect to the Gaction). A measurable subset A is called wandering if all its translations gA; g 2 G are mutually disjoint (mod 0 with respect to the measure m), and it is called recurrent S if A g6=e gA ; i.e., if for a.e. point ! 2 A there is g = g(!) 6= e such that g! 2 A. The action is called conservative if it admits no non-trivial wandering sets, and it is called completely dissipative if there exists a wandering set A (a \fundamental domain") such that the union of its translations gA; g 2 G is the whole space (mod 0).

If A is a wandering set, then for a.e. ! 2 A the orbit G! is an ergodic component of the G-action, and the group G acts freely on this orbit (an orbit with these two

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properties is a dissipative orbit ). Conversely, the restriction of a G-action onto any measurable set consisting of dissipative orbits is completely dissipative. Hence, the space admits a unique Hopf decomposition into the union of two G-invariant disjoint measurable sets C and D (conservative and dissipative parts of , respectively) such that the restriction of the action onto C is conservative, and the restriction onto D (which is the union of all dissipative orbits) is completely dissipative. An ergodic action is conservative unless the space ( ; m) consists of a single dissipative orbit. If m( ) < 1, then any measure preserving action of an in nite group G on the space ( ; m) is conservative. Proposition 1.1. The conservative part C of a measure type preserving G-action on a measure space ( ; m) is the maximal (mod 0) set in such that all its subsets are recurrent. S

Proof. Let A be a non-recurrent set, then A0 = A n g6=e gA is a non-empty wandering set, so that A has a non-trivial intersection with the dissipative part D. Conversely, as it follows from the de nition of the dissipative part D, if for a set A

the intersection A \ D is non-empty, then there is a wandering subset A0 A \ D.

If A is a recurrent set, then A \ E is also recurrent for any G-invariant set E , in particular, A \ D D is recurrent. On the other hand, for a measure preserving action of an in nite group G a subset A0 of its dissipative part can be recurrent only if m(A0 ) = 1. Thus, we have Proposition 1.2. For a measure preserving action of an in nite group G on a measure space ( ; m) a set A with m(A) < 1 is recurrent if and only if AS C . In particular, if there is a family of recurrent sets An such that m(An ) < 1 and An = , then the action is conservative. 1.1.2. Z-actions Any action of the group of integers Z is generated by a single invertible transformation T , so that instead of speaking about conservativity or dissipativity of a Z-action one can speak about conservativity or dissipativity of its generator T . Since the group Z = fT n g is linearly ordered, one can introduce the following notion: De nition 1.2. Let T be a (not necessarily invertible) measure type preserving transformation of a measure spaceS( ; m). A set A is (+)-recurrent with respect to the transformation T if A n 0 : T k ! 2 Ag : The induced transformation of a (+)-recurrent set A is de ned as TA (!) = T n(!) ! : If the transformation T is invertible, then a set A is called (?)-recurrent if it is (+)-recurrent with respect to the inverse transformation T ?1 .

A measure preserving transformation T of a measure space ( ; m) is called an endomorphism of this space (resp, automorphsim , if T is invertible). If A is a (+)-recurrent set of an endomorphism T , then TA is the induced endomorphism of the space (A; mA) (resp, induced automorphism if T is an automorphism).


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If T is invertible, and A is not a (+)-recurrent set, then A0 = A n n 0 determines the corresponding geodesic current on @ 2Xe uniquely (up to a

constant multiplier).

2.4. Applications

2.4.1. Conformal densities Let Xe be a covering hyperbolic space satisfying axiom (C). A family of nite measures fx : x 2 Xe g on the boundary @ Xe is called a conformal density of dimension if

dx1 ( ) = e? (x1 ;x2) 8 x ; x 2 X; e 2 @X e ; 1 2 dx2 where is the Busemann cocycle of the point [K1]. A conformal density fx g is called G-invariant if gx = gx 8 2 G; x 2 Xe :

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If fx g and f0x g are two conformal densities with the same dimension , then the Radon measure d( ?; +) = e2( ?j + )x dx ( ? )dx ( +) is independent of x by Proposition 2.2, hence it is G-invariant. Theorem 2.6 readily applies to the geodesic current . Thus, we have Theorem 2.7. Let Xe be a covering hyperbolic space satisfying axioms (U) and (C), and fx g { a G-invariant conformal density of dimension . Denote by [] the common type of measures x . Then the following dichotomy holds true. Either (1) The measure type [] is concentrated on the radial limit set r ; (2) The product measure type [] [] on @ 2 Xe is ergodic; or: (10) The measure type [] is concentrated on the complement r of the radial limit set r ; 0 (2 ) The product measure type [] [] on @ 2Xe is completely dissipative with respect to the G-action. If there is a conformal density fx g with x ( r ) > 0, then there are no other conformal densities with the same dimension (up to a constant multiplier), and fx g is concentrated on r . 2.4.2. Convex cocompact groups

Let now Xe be a general covering hyperbolic space (not necessarily with unique or convergent geodesics), and @ Xe be the limit set of the deck group G, i.e., the closure in @ Xe of the orbit Go of a reference point o 2 Xe (this de nition does not depend on the choice of o). The convex hull conv Xe of is the union of all geodesics in Xe with endpoints from . The group G is called convex cocompact in Xe if the action of G on conv is cocompact. If the group G is convex cocompact, then = r , the group G is word hyperbolic (i.e., G is a hyperbolic space with respect to a word metric on it), and = r can be identi ed with the hyperbolic boundary @G [Co]. Furthermore, one can construct a covering hyperbolic space Xe 0 with a cocompact G-action satisfying axioms (U) and (C) such that there is a G-equivariant homeomorphism between @ Xe and @ Xe 0 which extends to a homeomorphism between @ 2Xe and @ 2Xe 0 [Gr]. Thus, although in this situation one can de ne only a \quasi-geodesic" ow on Xe , the results of Theorem 2.6 concerning properties of quasi-product geodesic currents are still true. Theorem 2.8. Let Xe be a covering hyperbolic space with a properly discontinuous action of a convex cocompact group G (e.g., G is a word hyperbolic group, and Xe = G), and @ Xe { the limit set of G. If ? + is a G-invariant Radon measure on @ 2Xe , then either (1) ? ( ) = + ( ) = 1; (2) The measure on @ 2Xe is G-ergodic;


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or: (10) ? ( ) = + ( ) = 0; (20) The measure on @ 2Xe is completely dissipative with respect to the G-action;

2.5. Hyperbolicity and visibility A simply connected Riemannian manifold with non-positive sectional curvature is called a Cartan{Hadamard manifold [BGS]. In particular, the universal covering space of any non-positively curved Riemannian manifold is a Cartan{Hadamard manifold. Using the exponential map the space G X of in nite geodesic on X can be identi ed with the unit tangent bundle SX , so that the geodesic ow on G X in the sense of De nition 2.2 coincides with the usual geodesic ow on SX . Let @X be the space of asymptotic classes of geodesic rays on a Cartan{Hadamard manifold X . Then for any class 2 @X and any x 2 X there exists a unique geodesic ray starting from x and belonging to the class . Furthermore, any two dierent geodesic rays issued from the same point x belong to dierent asymptotic classes. Thus, for any reference point o 2 X the space of asymptotic classes @X can be identi ed with the unit sphere So of the tangent space To . This is why the space @X is called the sphere at in nity of the manifold X . The sphere at in nity @X is the boundary of the manifold X in the visibility compacti cation : a (tending to in nity) sequence xn 2 X is convergent in X = X \ @X if and only if for a certain reference point o 2 X directions of the vectors (expo )?1 (xn) 2 Tx converge (this compacti cation does not depend on the reference point o). The topology on @X is de ned by the cone neighbourhoods

Cx;" ( ) = f 0 2 @X : \x ( ; 0) < "g ; where \o ( ; 0) is the angle between the directing vectors of the geodesic rays [o; ] and [o; 0], i.e., the angle between the points and 0 as seen from the point o 2 X . De nition 2.6. A Cartan{Hadamard manifold X has uniform visibility property if for any > 0 there exists a number R = R() > 0 such that for any three points x; y; z 2 X ?

dist x; [y; z] > R ) \x (y; z) < ; ?

where dist x; [y; z] is the distance between the point x and the geodesic segment [y; z], and \x (y; z) is the angle between points y and z as seen from x, i.e., the angle between directing vectors of the geodesic segments [x; y] and [x; z]. In other words, uniform visibility means that the angle under which a geodesic segment [y; z] is seen from a xed point x tends to zero uniformly as the distance to this segment tends to in nity. Note that the uniform visibility for nite geodesic segments implies the uniform visibility also for in nite (one-sided or two-sided) geodesics.

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Remark. A weaker visibility property (when the constant R = R() may depend on x) is equivalent to existence of a geodesic joining any two distinct points ? ; 2 @X . +

See [EON] for a discussion of the relations between visibility and properties (C) and (U) for general Cartan{Hadamard manifolds.

For the sake of completeness we shall give here a proof of the following elementary statement connecting uniform visibility and hyperbolicity. The idea of this proof is due to W.Ballmann (see also [Ca], [Ki]). Theorem 2.9. A Cartan{Hadamard manifold X is hyperbolic with respect to the Riemannian metric if and only if it is has the uniform visibility property. Proof. Visibility ) hyperbolicity. Let x; y; z be three distinct points from X . As it follows from comparison with the

at case, the sum of the angles of the geodesic triangle (x; y; z) does not exceed , so that there are at least two angles which do not exceed =2. Let these be angles opposite to the point x. Then the basepoint p of the perpendicular [x; p] from x to the in nite geodesic determined by the points y and z lies between y and z (for, if p is outside of the segment [y; z], and y is the endpoint of the interval [y; z] nearest to p, then the triangle (x; y; p) has two angles not less than =2 at vertices p and y). Now, the uniform visibility axiom implies that dist(p; [x; y]) and dist(p; [x; z]) both do not exceed R(=2). Thus, there is a triangle with vertices on sides of the original triangle (x; y; z) and diameter not more than 2R(=2). By Proposition 2.1(b) this implies that X is -hyperbolic with the constant = 8R(=2). Hyperbolicity ) visibility. Consider a geodesic triangle (o; x1; x2) with angle at the vertex o. Let R = dist(o; [x1; x2]), and d = (x1 jx2)o . Then by Proposition 2.1(a)

R d R ? 4 Let yi 2 [o; xi] (i = 1; 2); z 2 [x1; x2] be the vertices of the inscribed triangle of the triangle (o; x1; x2), so that dist(o; x1) = dist(o; x2) = d, and diam (y1; y2; z) 4 (Proposition 2.1(a)). Then, as it follows from comparison with the at case, 4 dist(y1 ; y2) 2d sin(=2) : Comparison of this inequality with the preceding one gives 4 2(R ? 4) sin(=2) ; or

2 R R() = 4 + sin(=2) : Thus, X is a uniform visibility manifold with the constant R().


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Theorem 2.10. If X is a hyperbolic ( uniform visibility) Cartan{Hadamard manifold

with uniformly bounded sectional curvatures, then the uniform structures on @X (and corresponding topologies on @ 2X ) determined by hyperbolic and cone neighbourhoods coincide.

In the proof of Theorem 2.10 we shall need the following geometrical result. Proposition 2.4. Let X be a hyperbolic Cartan{Hadamard manifold. Take a reference point o 2 X and two points 1 6= 2 2 @X . Let xi 2 [o; i] (i = 1; 2) and z 2 [ 1; 2] be the vertices of the inscribed triangle of the geodesic triangle (o; 1; 2), so that jx1 j = jx2j and i (xi ; z) = 0. Denote by yi 2 [o; i] the points uniquely determined by the conditions dist(o; y1) = dist(o; y2) and dist(y1 ; y2) = 1. Put

p = po ( 1; 2) = dist(o; xi) = ( 1j 2)o and

l = lo ( 1; 2) = dist(o; yi ) :

Then

n o min p; 4p l p + 4 + 21 : Proof. We shall consider two cases depending on the sign of the dierence l ? p. Case 1 (l < p). It means that the points yi lie on the sides of the geodesic traingle (o; x1; x2). Since dist(x1 ; x2) 4 by Proposition 2.1(a), comparison with the at case by the Alexandrov comparison theorem [Al], [BGS] immediately gives the inequality l p=(4) : Case 2 (l > p). Denote by zi the points on the geodesic [ 1; 2] determined by the relations i (yi; zi ) = 0. Then by Proposition 2.1(a)

dist(yi ; zi ) dist(xi; z) 4 ; and dist(z1; z2 ) = dist(z; z1 ) + dist(z; z2) = dist(x1 ; y1) + dist(x2; y2) = 2(l ? p) ; so that 2(l ? p) = dist(z1 ; z2) dist(z1; y1) + dist(y1; y2) + dist(y2 ; z2) 8 + 1 ;

whence l ? p 4=d + 21 :

Remark. In the case when the sectional curvatures of X are pinched the dierence l ? p is in fact uniformly bounded (see [K1, Proposition 1.4]). Proof of Theorem 2.10. Let 8
0 there is a constant C = C (D ) with the property that 1 G(x; y) C 8 x; y 2 X : dist(x; y) D ;

and

( D1 )

(D2 )

C

G(x1 ; x3) 1 C G(x1 ; x2)G(x2; x3) C whenever there exist a geodesic segment : I ! X and numbers t1 ; t2; t3 2 I such that t1 t2 t3 and ? dist xi ; (ti) D i = 1; 2; 3 : In particular, G(x; y) > 0 for all x; y 2 X , and the operator P is irreducible. There exists a function : R + ! R + such that (t) ! 0 as t ! 1, and ? G(x; y)G(y; x) dist(x; y) 8 x; y 2 X : There exists a constant C > 0 such that X npn (x; x) C 8 x 2 X : n1

(D3 )

There exists a function : R + ! R + such that Ct 8C > 0 ; tlim !1 e (t) = 0 and

(G)

?

p(x; y) dist(x; y) 8 x; y 2 X : There exist constants K and v such that for any point x 2 X and any R > 0 the measure m of the R-ball centered at x satis es the inequality ? m BR (x) KeRv :


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Here condition (M) means that the Green kernel is almost multiplicative along all geodesics, conditions (D1), (D2 ), (D3) guarantee a suciently fast decay of the transition probabilities and of the Green kernel in various senses, and condition (G) says that the rate of (exponential) growth of the space X with respect to the measure m is uniformly bounded. Note that condition (M) immediately implies the following property of the Green function: (M1 ) There are constants C1 ; C2 such that G(x; y) ; G(y; x) C eC2 dist(y;z) 8 x; y; z 2 X : G(x; z) G(z; x) 1 Since

Z

pk (x0; x)pn(x; x0) dm(x) = pk+n (x0 ; x0)

8 k; n > 0; x0 2 X ;

condition (D2 ) implies that (D4 ) There exists a constant C > 0 such that Z

G(x0 ; x)G(x; x0) dm(x) C

8 x0 2 X :

Remarks. 1. Condition (D ) would follow from a stronger condition r(P ) < 1, where 2

r(P ) is the spectral radius of P as an operator in the space L2 (X; m). 2.Conditions (D3 ) and (G) are always satis ed for diusion processes on Riemannian manifolds with bounded geometry (where m is the Riemannian volume) and for nite range random walks on graphs with uniformly bounded degrees of vertices (where m is uniformly equivalent to the counting measure). 3. Condition (D1) can be deduced from r(P ) < 1 and conditions (D3) and (G) (cf. [KW]). 4. The most important here is condition (M). Under the assumption r(P ) < 1 it was proved by Ancona [An1], [An2] for diusion processes on Riemannian manifolds with bounded geometry which are hyperbolic as metric spaces and for irreducible nite range random walks on Rhyperbolic graphs. [Ancona de nes the Green kernel for diusion 1 processes as G(x; y) = o pt (x; y) dt, where pt (; ) are time t transition densities of the diusion process,Pbut his results can be easily carried over to the discrete time Green kernel G(x; y) = 1 n=1 pn (x; y ).] 5. Recently, Cao [Ca] and Kifer [Ki] have independently proved that for a Cartan{ Hadamard manifold M its hyperbolicity (as a metric space) implies that the bottom of the spectrum of the Laplacian M is non-zero, i.e., r(P ) < 1 for the time 1 transition operator P of the Brownian motion on M . Thus, conditions (M), (D1), (D2), (D3), (G) are satis ed for the Brownian motion on any Cartan-Hadamard manifold with the uniform visibility property and uniformly bounded sectional curvatures. Below we shall use the notation A 4 B if the ratio A=B is uniformly bounded from above by an absolute constant, and the notation A B if A 4 B and B 4 A, i.e., if the ratio A=B is uniformly bounded both from below and from above.

34


3.1.2. The Martin boundary R Let Pf (x) = f (y)p(x; y) dm(y) be a Markov operator with densities p(x; y) with respect to a Radon measure m on a locally compact topological space X , and nite Green kernel X G(x; y) = pn (x; y) : n1

Denote by

(x; y) Ky (x) = G G(o; y)

the Martin kernel of the operator P , where o 2 X is a xed reference point. Then y 7! Ky () is an embedding of the space X into the space F (X; R+ ) of positive functions on X endowed with pointwise convergence topology. The Martin boundary @M X = @M (X; P ) is the set of all limit points of the Martin kernels fKy g in F (X; R+ ) as y tends to in nity in X . If the set fKy : y 2 X g is relatively compact (i.e., the functions Ky are pointwise uniformly bounded), then the union X [ @M X is a compacti cation of the space X which is called its Martin compacti cation [Dy], [Br]. All minimal positive harmonic functions of the operator P belong to the Martin boundary (the corresponding subset of the Martin boundary is called the minimal Martin boundary ), and for any positive harmonic function f there exists a unique representing measure f concentrated on the minimal Martin boundary such that

f (x) =

Z

!(x) df (!)

8x 2 X :

The Martin boundary is connected with the Poisson boundary in the following way:

m PZ+ -a.e. path fxn g on X converges in the Martin topology to a point x1 2 @M X ,

and the boundary @M X considered as a measure space with the corresponding hitting distributions (harmonic measures) is isomorphic to the Poisson boundary of the operator P . The harmonic measure o of the reference point o coincides with the representing measure 1 of the constant function 1(x) 1, and for a.e. ! 2 @M X

dx (!) : !(x) = xlim K ( x ) = x n d n !! o

3.1.3. The Martin boundary of hyperbolic spaces Theorem 3.1. Let P : L1 (X; m) - be a Markov operator on a hyperbolic space X satisfying conditions (M) and (D1). Then the Martin boundary @M X of the space X corresponding to the operator P coincides with the hyperbolic boundary of X .

Corollary. For mPZ+-a.e. path fxn g there exists the limit lim x = x1 2 @X ; n!1 n and the hyperbolic boundary @X with the corresponding family of hitting distributions is isomorphic to the Poisson boundary of the operator P . The harmonic measures


35

x ; x 2 X on @X are all equivalent and their Radon-Nikodym derivatives satisfy the inequality

dx ( ) C eC2 dist(y;z) 1 dy

8x; y 2 X; 2 @X ;

where C1 ; C2 are constants from condition (M1 ).

Our proof of Theorem 3.1 essentially mimics Ancona's proof from [An1] in slightly different terms { we make an emphasis here on using the almost multiplicativity condition (M). First we shall prove the following Lemma. Lemma 3.1. Under condition (M) for any two sequences (xn ) and (x0n ) converging to a point 2 @X in the hyperbolic topology

Kxn (x) " lim inf n!1 K 0 (x) xn

8x 2 X ;

where " > 0 is a constant depending only on the hyperbolicity constant and the constants from condition (M). Proof. Let be a geodesic ray joining o and , and n { a geodesic segment joining o and xn . Then by Proposition 2.1(a) applied to the geodesic triangle (o; xn; ), ?

dist (tn ); n(tn ) 4 ;

tn = (xn j ) ;

so that by Proposition 2.1(d) ?

dist (t); n(t) 4 + 8 = 12

8 t 2 [0; tn] :

Thus, by (M),

G(o; xn ) G(o; y)G(y; xn)

8 y 2 : dist(o; y) (xn j ) :

Replacing o by an arbitrary point x gives that for any geodesic ray joining x and

G(x; xn ) G(x; z)G(z; xn )

8 z 2 : dist(x; z) (xn j )x :

Since (xn j ) ! 1, and the rays [o; ] and [x; ] are asymptotic, by Proposition 2.1(e) the points y and z can be chosen in such a way that dist(y; z) 16. Replacing xn by x0n and applying (M) gives

Kxn (x) = G(x; xn )G(o; x0n) Kx0n (x) G(x; x0n )G(o; xn) G(x; z)G(z; xn )G(o; y)G(y; x0n) = G(z; xn ) G(y; x0n) 1 : G (x; z)G(z; x0n )G(o; y)G(y; xn) G(y; xn) G(z; x0n )

Proof of Theorem 3.1. The proof consists of two parts.

36


First we shall show that any sequence (xn ) in X which converges in the hyperbolic topology to a point 2 @X converges also in the Martin topology. Denote by C @M X the set of all Martin limit points of sequences (xn ) which converge to in the hyperbolic topology, and let n

F = f : f =

Z

! d(!); (C ) = (@M X ) = 1

o

be the convex set of normalized positive functions which admit an integral representation supported by C . As it follows from Lemma 3.1, there is a constant " > 0 such that !1 "!2 8 !1; !2 2 F : Being positive, the normalized dierence (!1 ? "!2 )=(1 ? ") also belongs to F , so that by iteration we have that for any n 0 !1 "!2 + "(1 ? ")!2 + + "(1 ? ")n !2 ; which implies that !1 !2 . Thus, F (and, as a consequence, C ) consists of a single point. Now we have to show that if two sequences (xn ) and (x0n ) converge to two dierent points ; 0 2 @X , then their Martin limit points are also dierent. Only at this point we shall need condition (D1 ). As it follows from the rst part of the proof, it is sucient to take any two xed sequences xn ! and x0n ! 0. In particular, we can assume that all points xn and x0n lie on an in nite geodesic joining 0 and . Denote by o0 2 the projection of the reference point o onto (see x2.1), i.e., the corresponding vertex of the inscribed triangle of the geodesic triangle (o; ; 0). Then G(o; y) G(o; o0 )G(o0 ; y) uniformly for any y 2 as it follows from Proposition 2.1(a) and condition (M) (cf. proof of Lemma 3.1). Let us parametrize the geodesic in such a way that (0) = o0 , and put xn = (n); x0n = (?n). Then condition (D1) implies that for any x = (t); t 0 Kxn (x) = G(x; xn)G(o; x0n ) G(x; xn)G(o; o0 )G(o0 ; x0n ) = G(x; xn )G(o0 ; x0n) Kx0n (x) G(o; xn)G(x; x0n ) G(o; o0 )G(o0 ; xn)G(x; x0n ) G(o0 ; xn )G(x; x0n) 1 (x; xn)G(o0 ; x0n ) ? G(o0 ; x)G 0 0 0 !1 1 ; G(x; xn)G(x; o )G(o ; xn) dist(o0 ; x) t?! so that the Martin limit points of the sequences (xn) and (x0n ) must be dierent.

Remark. In fact, the second part of this proof shows that the boundary of the Con-

stantinescu-Cornea compacti cation [Br] of a hyperbolic space X obtained from any kernel satisfying conditions (M) and (D1 ) is a bration over the hyperbolic boundary of X.

3.2. Construction of the harmonic invariant measure

3.2.1. An analogue of the Hall Lemma Recall that the R-shadow Sx (y; R) @X of a point y 2 X with respect to a point x 2 X is the set of endpoints of all geodesic rays starting from a point x 2 X and such that dist(x; ) R (x2.3.1).


37

Lemma 3.2. Under condition (M) there exists a constant C = C (R) > 0 such that ? x Sx (y; R) CG(x; y) 8 x; y 2 X : Proof. Let 2 Sx (y; R), and xn ! along a geodesic ray joining x and . Then

dx ( ) = lim Kxn (x) = lim G(x; xn) G(x; y) n!1 Kxn (y ) n!1 G(y; xn ) dy by condition (M). On the other hand, Z ? dx ( ) d ( ) supn dx ( ) : 2 S (y; R)o : x Sx (y; R) = y x dy Sx (y;R) dy

Corollary (cf. [KL]). Under conditions (M) and (D ) the harmonic meaasure class [ ] 1

on the hyperbolic boundary @X does not have atoms.

Proof. Take a geodesic ray joining x and . Since 2 Sx (y; R) for any point y 2 and any R > 0, we have ? x ( ) x Sx (y; R) CG(x; y) ; which by condition (D1) implies that x ( ) = 0.

Remark. As it follows from condition (M), G(x; y) is uniformly equivalent to the prob-

ability of ever ? hitting the ball B (y; R) starting from the point x. On the other hand, x Sx (y; R) is the probability of hitting (at in nity) the radial projection of this ball onto @X . Thus, Lemma 3.2 gives an upper estimate of the probability of hitting the radial projection of the ball B (y; R) in terms of the ? probability of hitting the ball B (y; R) itself (in order to hit the shadow x Sx (y; R) the random path has to pass through B (y; R) with a bounded away from zero probability). So, Lemma 3.2 is an analogue of the Hall Lemma well known in the harmonic analysis (e.g. [Dur], see also [LMT] for generalizations to rank one symmetric spaces). 3.2.2. An estimate for products of harmonic functions If m is a stationary measure of the operator P , then the Green function G of the reversed operator P satis es the relation G (x; y) = G(y; x) 8 x; y 2 X ; and conditions (M), (G), (D1 ), (D2), (D3) are also satis ed for the reversed operator. In particular, its Martin boundary can be also identi ed with the hyperbolic boundary @X . Denote by x+ and x? the harmonic measures of the forward operator P and the backward operator P , respectively, and by

' (x) = dx ( ) ; x 2 X do their minimal harmonic functions corresponding to a point 2 @X and normalized by the condition ' (o) = 1.

38


Lemma 3.3. Let x 2 X and ? 6= + 2 @X . Take a geodesic with endpoints ? ; +, and denote by o0 the projection of the reference point o onto . Then under condition (M) there exist constants K1 ; K2 such that

?

'? ? (x)'+ + (x) K1 G(x; o0 )G(o0 ; x) exp K2 j ?; + (o; x)j + ( ? j +)

8x 2 X ;

where ? ; + is the cocycle on X introduced in x2.1.2. Proof. Parametrize the geodesic in such a way that (0) = o0 and (t) ! + as t ! 1. Then G(x; (t)) '+ + (x) = tlim !1 G(o; (t)) and G((?t); x) ; '? ? (x) = tlim !1 G((?t); o) so that by Proposition 2.1(a) and condition (M) 0 (x0 ; x) 1 0 )G(o0 ; x) G(x; x0) G(x0 ; x) = G ( x; o '? ? (x)'+ + (x) GG((x;o; xo0 ))G 0 0 0 0 G(o ; o) G(x; o ) G(o ; x) G(o; o )G(o0 ; o) :

Since dist(x0 ; o0 ) = j ?; + (x)j ; and jdist(o; o0) ? ( ? j +)j 4 the claim follows from condition (M1 ). 3.2.3. The harmonic invariant measure Theorem 3.2. Let P : L1 (X; m) - be a Markov operator on a hyperbolic space X with an invariant measure m satisfying condition (M), (G), (D1), (D2 ), (D3), and let m PZ be the shift invariant measure in the space of bilateral paths X Z with one-dimensional stationary distribution m. Take a subset A0 X Z of the path space which is wandering with respect to the time shift T and such that

(1)

XZ =

[

n2Z

T n A0

(mPZ-mod 0) :

Let bnd ? bnd + be the map assigning to a path fxn g the pair of its boundary points (bnd ? fxn g; bnd + fxn g) 2 @ 2 X , and denote by the bnd ? bnd + -image of the restriction of the measure m PZ onto A0. Then the measure does not depend on the choice of A0 and is a Radon measure on @ 2X . Proof. First of all note that the operator P : L1 (X; m) - is transient, as it follows from niteness of its Green kernel, so that the time shift in the path space (X Z; mPZ) is completely dissipative (Theorem 1.2). Thus, the decomposition (1) exists, and the measure does not depend on the choice of A0 . The only thing we have to show is that the measure is Radon (cf. proof of Theorem 2.1). Consider the decomposition m PZ =

Z Z

? + m PZ

? ; + dx ( ? ) dx ( + ) ;


39

of the measure mPZ with respect to the forward and backward Poisson boundaries of the operator P (see Proposition 1.6). For a path fxn g with limit points = limn!1 xn 2 @X (which exist a.e. by Theorem 3.1) let fxn g = minfn 2 Z : xn 2 A+ ? ; + g ; where A+ ?; + = fx 2 X : ?; + (o; x) 0g ; and ? ; + (o; x) = (xj +) ? (xj ?) (see x2.1.2). Then the set A0 = fxn g : fxng = 1 is wandering with respect to the shift in the path space, and the union of its translations is the whole path space, which shows that the measure is absolutely continuous with respect to x? x+ , and d ( ; ) = PZ fx g = 1 : dx? x+ ? + m ? ; + n Fix ? 6= + 2 @X , and put A = A ? ; + (since both measures + and ? do not have atoms, we can always assume ? 6= + ). Then Z + + m PZ

? ; + fxn g = 1 Zm P ? ; + x0 62 A ; x1 2 A Z+ x1 2 A+ '? (x)'+ (x) dm(x) : P = x

+

?

+ A?

As it follows from Theorem 3.1 and conditions (G) and (D3), there exists a function 0 : R + ! R+ such that lim eCt 0 (t) = 0

t!1

and

8C > 0 :

+ 0? + 0? x PZ

++ x1 2 A dist(x; A ) jD ? ; + (o; x)j

Thus, by Lemma 3.3,

? + + x PZ

++ x1 2 A ' ? (x)' + (x) K1 supfeK2 t 0 (t) : t 0g eK2 ( ? j +) G(x; o0 )G(o0 ; x) :

Integrating this inequality by dm(x) gives by condition (D4 ) the desired uniform bound of the derivative d ( ; ) ? dx x+ ? + in terms of the Gromov product ( ?j + ). By Theorem 2.1, if the space X satis es condition (U), the measure de nes a Radon invariant measure of the geodesic ow on G X which is called the harmonic invariant

40


measure . If Xe is a covering hyperbolic space with the deck transformations group G e , and Pe : L1 (X; e m and the quotient space X = X=G e ) - is a covering Markov operator, then the measure is obviously G-invariant, and determines the harmonic invariant measure of the quotient geodesic ow. In this case we shall say that is the harmonic geodesic current corresponding to the operator Pe. Ledrappier [Le] proposed an ergodic construction of a harmonic invariant measure for the geodesic ow on compact negatively curved manifolds (corresponding to the Brownian motion on its universal covering space), and proved its ergodicity (in fact, he showed that the harmonic invariant measure is a Gibbs measure in this case). Later another, more geometric construction (but also essentially using the compactness assumption) was given by Hamenstadt [Ha]. The compactness assumption was dropped out in [K1], where a harmonic geodesic current was constructed for an arbitrary negatively curved manifold M with pinched sectional curvatures. The exposition in this Section goes essentially along the same lines as in [K1]. Another way of obtaining a geodesic current from the harmonic measure class [ ? ]

[ + ] on @ 2Xe is to choose a density '( ? ; +) such that the measure

d( ?; +) = '( ?; +)do? ( ?)do+ ( +) on @ 2Xe is a G-invariant Radon measure. For example, in our situation one can take for ' the so-called Naim kernel [Ko]

K + (x) = lim inf lim G(x; y) : ( ? ; +) = lim inf x! ? G(x; o) x! ? y! + G(x; o)G(o; y ) The resulting measure on @ 2Xe is clearly G-invariant, and it is Radon by property (M) (see also [A2]). One could also, for example, replace lim inf in the de nition of by lim sup. As it follows from Theorem 1.4, in the case when the quotient operator P is recurrent, the measure class [ ? ] [ +] is G-ergodic, so that the measures and must be proportional. Apparently, this is not necessarily the case in the general situation. Note that one could ask a purely measure theoretical question about existence of a G-invariant measure on the measure class [ ? ] [ + ] for a general covering Markov operator. Unfortunately, we were not able to deal with this question using measure theoretical means only, and we had to involve here the Martin boundary considerations.

3.3. The main theorem and its applications 3.3.1. The main theorem e m Theorem 3.3. Let Xe be a covering hyperbolic space, Pe : L1 (X; e ) - { a covering 1 e Markov operator on X with stationary measure m e , and P : L (X; m) - { the corresponding quotient operator on the quotient space (X; m). Suppose that the hyperbolic e m space Xe satis es condition (U), and the operator Pe : L1 (X; e ) - satis es conditions + e of the (M), (G), (D1 ), (D2 ), (D3 ). Denote by [ ] the harmonic measure type on @ X operator Pe, and by [ ? ] the harmonic measure type of the backward operator Pe. Let


41

N be the harmonic invariant measure of the quotient geodesic ow on G X . Then the following dichotomy holds true. Either: (1) The quotient operator P : L1 (X; m) is recurrent; P (2) The sum g2G G(x; gy) is in nite for a.e. x; y 2 Xe , where G is the Green e m function of the covering operator Pe : L1 (X; e) - ; f; (3) Both measure classes [ ] and [ ] are supported by the radial limit set r @ M t (4) The quotient geodesic ow fT g on G X is ergodic with respect to the harmonic invariant measure N; (5) The quotient geodesic ow fT t g on G X is conservative with respect to the harmonic invariant measure N; (6) The action of the group G on @ 2Xe is ergodic with respect to the measure type [ ] [ ]; (7) The action of the group G on @ 2Xe is conservative with respect to the measure type [ ] [ ]. Or: (10) The quotient operator P : L1 (X; m) is transient; P (20) The sum g2G G(x; gy) is nite for a.e. x; y 2 Xe ; (30) Both measure classes [ ] and [ ] are supported by the complement r of the radial limit set; (40) The quotient geodesic ow fT t g on G X is non-ergodic with respect to the harmonic invariant measure N; 0 (5 ) The quotient geodesic ow fT t g on G X is completely dissipative with respect to the harmonic invariant measure N; 0 (6 ) The action of the group G on @ 2Xe is non-ergodic with respect to the measure type [ ] [ ]; (70) The action of the group G on @ 2Xe is completely dissipative with respect to the measure type [ ] [ ]. Proof. First of all note that as it follows from condition (M), the quotient P operator P : 1 L (X; m) is irreducible, so that it is either recurrent, or transient. Since g2G G(x; gy) coincides with the Green kernel of the quotient operator P : L1 (X; m) - , one immediately has equivalences (1) () (2) and (10) () (20). If P is recurrent, then Theorem 1.4 gives (6) and (7), which by Theorem 2.2 imply (4) and (5), and Theorem 2.4 yields (3). P Now suppose that P is transient, and g2G G(x; gy) < 1. Since the radial limit set

r can be presented as [ \ [

r = Sx (gi y; R) R>0 n1 in

(see x2.3), Lemma 3.2 implies that the harmonic measure types [ + ] (and [ ? ]) are concentrated on r . Thus, by Theorem 2.4, the measure type [ ] [ ] and the measure N are completely dissipative. Since there are no atoms in [ ] [ ] (Corollary of Lemma 3.2), this implies ergodicity of the measure type [ ] [ ] and the measure N.

42


Remark. Note that the proof of ergodicity of the quotient geodesic ow on G X does

not use the general Theorem 2.5, and is based on results from Section 1 on covering Markov operators. Thus, our proof of Theorem 3.3 is obtained without any use of the Birkho ergodic theorem (and even the von Neumann ergodic theorem, because Theorem 1.4 can be proved by purely Markov means). 3.3.2. Cartan{Hadamard manifolds Let Xe be a regular cover of a Riemannian manifold X (in particular, the universal covering space). Then any geodesic on X can be uniquely (up to a translation) lifted to Xe , so that the space G X can be identi ed with the unit tangent bundle SX , and the quotient geodesic ow on G X coincides with the usual geodesic ow on SX . In view of the discussion in x3.1.1, assumptions of Theorem 3.3 are satis ed when X is a non-positively curved manifold with bounded geometry, and its universal covering space Xe has the uniform visibility property and property (U), me is the Riemannian volume on Xe , and Pe is the time one Markov operator of the Brownian motion on Xe . In this situation the operator Pe is reversible, so that the mesure types [ ? ] and [ + ] coincide. Note that for Cartan{Hadamard manifolds with pinched sectional curvature equivalence of conditions (1), (3), (5), (6) on one side, and conditions (10 ), (50 ) on the other side was proved by Ancona [An2] (in slightly dierent terms) by using potential theory methods. If Xe is a negatively curved symmetric space of rank one, then the harmonic measure type on the boundary @ Xe coincides with the visibility (the Lebesgue) measure type, and the harmonic invariant measure coincides with the Liouville invariant measure of the geodesic ow. Further, in this case the Green function is asymptotically exponential, i.e., G(x; y) e?vdist(x;y) ; dist(x; y) 1 ; where 1 log vol B (x; R) v = Rlim !1

R is the exponential rate of growth of Xe (e.g., see [Y]). Thus, as a corollary of Theorem 3.3 we obtain the corresponding dichotomy for the Liouville invariant measure of the geodesic ow on locally symmetric rank 1 spaces . In the constant curvature case it is known as the Hopf{Tsuji{Sullivan theorem [Su2], [N]. Another possible application of Theorem 3.3 is to the invariant measure of the geodesic ow arising from a conformal density of dimension d=2 of a discontinuous groups of motions of the hyperbolic space H d+1 (cf. [Su3]). Theorem 3.3 gives a criterion of ergodicity of the geodesic ow with respect to the harmonic invariant measure. There are two other natural invariant measures of the geodesic ow on a negatively curved manifold { the Liouville invariant measure (corresponding to the visibility measure type on the sphere at in nity), and the \maximal entropy invariant measure" corresponding to a conformal density on the sphere at in nity. In the cocompact case the conformal density is unique and gives rise to the Bowen{Margulis invariant measure of the geodesic ow { see [K1]. In general all these


43

measures are dierent, so that one should not expect that ergodicity of the harmonic invariant measure might be connected with the ergodicity of the other two measures. It would be interesting to construct explicit examples of manifolds such that, for example, the geodesic ow is ergodic with respect to the harmonic invariant measure, but not ergodic with respect to the Liouville (or, the maximal entropy) invariant measure. [Recall that in the cocompact case ergodicity of all three measures follows from Theorem 2.6]. It would be also interesting to construct examples of this kind for universal covering trees (see below). Theorem 3.3 shows a striking similarity with analogous results about the conformal densities on Cartan{Hadamard manifolds with pinched sectional curvature [Y]: if is a conformal density of dimension on @ Xe , then the corresponding invariant measure of the geodesic ow is either completely dissipative or conservative (see Theorem 2.7), and P its conservativity is equivalent to convergence of the Poincare series g2G e?dist(x;gy) for all x; y 2 Xe . It would be interesting to be able to obtain both this result and Theorem 3.3 in a uni ed manner. 3.3.3. Trees Let X be a connected graph, and Xe { its universal covering tree. For the sake of simplicity assume that all edges of X has length 1 (in fact, this assumption is not relevant). Any path on X without \backtracking" (i.e., such that (n?1) 6= (n+1) 8 n) can be uniquely (up to a translation by the fundamental group G = 1(X )) lifted to a geodesic on Xe , thus, in this situation the quotient geodesic ow is the shift in the space of paths of X without backtracking. If degrees of vertices of X are uniformly bounded (this is an analogue of boundedness of geometry), then the Markov operator Pe of the simple random walk on Xe (with transition probabilities p(x; y) = 1= deg y whenever x and y are two neighbouring vertices) e has the property that is reversible with respect to the measure m e (x) = deg x. If X the lengths of all segments without branching are uniformly bounded (in particular, if X is nite and contains at least two dierent cycles), then the tree Xe satis es the strong isoperimetric inequality [Ge1], so that r(Pe) < 1 [Ge2], [K2] (multiplicativity of the Green function for trees can be veri ed directly [PW]). So, in this case one can de ne the harmonic invariant measure of the geodesic ow, and Theorem 3.3 gives a list of conditions equivalent to its ergodicity. For trees one can also de ne analogues of the Liouville invariant measure of the geodesic ow, and of the maximal entropy measure. The rst one corresponds to the \uniformly distributed geodesic ow" { the position of (n + 1) conditioned by (n) and (n ? 1) is equidistributed on the set of all neighbours of (n) except for (n ? 1). The corresponding geodesic current on @ 2Xe can be easily expressed in terms of the \visibility" measures on @ Xe (for a given point x 2 Xe the visibility measure !x is obtained from the \equidistributed measure" on the space of all geodesic rays emanating from x { at every vertex all branches directed outwards from x have the same probability). The second one corresponds to a conformal density on @ Xe (see x3.4.1). In general these three invariant measures of the geodesic ow are mutually singular (see [Ly]), but they coincide (up to a constant multiplier) in the case when X is a bihomogeneous graph , i.e., if degrees of vertices of X can take not more than two values q1

44


and q2 , and for any two neigbouring vertices one has degree q1 and the other has degree q2 . In this situation Theorem 3.3 gives a criterion of ergodicity of the geodesic ow with respect to any one of three (coinciding) invariant measures which is an analogue of the Hopf{Tsuji{Sullivan theorem (bi-homogeneous graphs being analogues of rank 1 locally symmetric spaces). Note that in the particular case of the invariant measure arising form a conformal density on the covering tree the dichotomy of Theorem 2.6 was proved in [CP]. References [Ah] L. V. Ahlfors, Mobius transformations in several dimensions, University of Minnesota, 1981. [Al] A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, in Russian, Trudy Steklov. Mat. Inst. 38 (1951), 5{23. [An1] A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math. 125 (1987), 495{536. [An2] A. Ancona, Theorie du potentiel sur les graphes et les varietes, Springer Lecture Notes in Math. 1427 (1990), 4{112. [Bo] F. Bonahon, Geodesic currents on negatively curved groups, Arboreal Group Theory (R. C. Alperin, ed.), MSRI Publ., vol. 19, Springer, New York, 1991, pp. 143{168. [BGS] W. Ballmann, M. Gromov, V. Schroeder, Manifolds of Nonpositive Curvature, Birkhauser, Basel, 1985. [Br] M. Brelot, On Topologies and Boundaries in Potential Theory, Springer, Berlin, 1971. [Ca] J. Cao, A new isoperimetric constant estimate and its application to the Martin boundary problem, preprint, 1993. [CDP] M. Coornaert, T. Delzant, A. Papadopoulos, Geometrie et theorie des groupes, Lecture Notes in Math., vol. 1441, Springer, Berlin, 1990. [CFS] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic Theory, Springer, New York, 1982. [Co] M. Coornaert, Sur les groupes proprement discontinus d'isometries des espaces hyperboliques au sens de Gromov, These, Universite de Strasbourg, 1990. [CP] M. Coornaert, A. Papadopoulos, Une dichotomie de Hopf pour les ots geodesiques associes aux groupes discrets d'isometries des arbres, Trans. Amer. Math. Soc. (to appear). [Dug] J. K. Dugdale, Kolmogorov automorphisms in - nite measure spaces, Publ. Math. Debrecen 14 (1967), 79{81. [Dur] P. L. Duren, Theory of H p Spaces, Academic Press, New York, 1970. [Dy] E. B. Dynkin, Markov Processes and Related Problems of Analysis, London Mathematical Society Lecture Note Series 54, Cambridge University Press, Cambridge, 1982. [EON] P. Eberlein, B. O'Neill, Visibility manifolds, Paci c J. Math. 46 (1973), 45{109. [Fo] S. R. Foguel, Harris operators, Israel J. Math. 33 (1979), 281{309. [Ga] L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983), 285-311. Akad. Wiss., Math.[Ge1] P. Gerl, Eine isoperimetrische Eigenschaft von Baumen, Sitzungsber. Ost. naturw. Klasse 195 (1986), 49{52. [Ge2] P. Gerl, Random walks on graphs with a strong isoperimetric inequality, J. Theor. Prob. 1 (1988), 171{187. [GH] Sur les Groupes Hyperboliques d'apres Mikhael Gromov (E. Ghys, P. de la Harpe, eds.), Birkhauser, Basel, 1990. [Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), MSRI Publ., vol. 8, Springer, New York, 1987, pp. 75{263. [Ha] U. Hamenstadt, An explicit description of harmonic measure, Math. Z. 205 (1990), 287{299. [Ho] E. Hopf, Ergodic theory and the geodesic ow on surfaces of constant negative curvature, Bull. Amer. Math. Soc. 77 (1971), 863{877. [HR] T. E. Harris, H. Robbins, Ergodic theory of Markov chains admitting an in nite invariant measure, Proc. Nat. Acad. Sci. USA 39 (1953), 860{864. [K1] V. A. Kaimanovich, Invariant measures of the geodesic ow and measures at in nity on negatively curved manifolds, Ann. Inst. H. Poincare, Physique Theorique 53 (1990), 361{393.


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[K2] V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Analysis 1 (1992), 61{82. [K3] V. A. Kaimanovich, Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy,, Proceedings of the International Conference on Harmonic Analysis and Discrete Potential Theory, Frascati, 1991 (M.A.Picardello, ed.), Plenum, New York, 1992, pp. 145{180. [K4] V. A. Kaimanovich, The Poisson boundary of covering Markov operators, preprint, 1993. [Ki] Y. Kifer, Spectrum, harmonic functions, and hyperbolic metric spaces, preprint, 1992. [KL] Y. Kifer, F. Ledrappier, Hausdor dimension of harmonic measures on negatively curved manifolds, Trans. AMS 318 (1990), 685{704. [Ko] T. Kori, La theorie des espaces fonctionnelles a nullite 1 et le probleme de Neumann sur les espaces harmoniques, Ann. Inst. Fourier Grenoble 27 (1977), 45{119. [Kr1] U. Krengel, Darstellungssatze fur Stromungen und Halbstromungen I, Math. Ann. 176 (1968/69), 181{190. [Kr2] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. [KS] U. Krengel, L. Sucheston, Note on shift-invariant sets, Ann. Math. Stat. 40 (1969), 694{696. [KW] V. A. Kaimanovich, W. Woess, The Dirichlet problem at in nity for random walks on grphs with a strong isoperimetric inequality, Prob. Th. Rel. Fields 91 (1992), 445-466. [Le] F. Ledrappier, Ergodic properties of the Brownian motion on covers of compact negatively curved manifolds, Bol. Soc. Bras. Mat. 19 (1988), 115{140. [Ly] R. Lyons, Equivalence of boundary measures of covering trees of nite graphs, Ergod. Th. & Dynam. Sys. (to appear). [LMT]T. J. Lyons, K. B. MacGibbon, J. C. Taylor, Projection theorems for hitting probabilities and a theorem of Littlewood, J. Funct. Anal. 59 (1984), 470{489. [N] P. J. Nicholls, Ergodic Theory of Discrete Groups, Cambridge Univ. Press, 1989. [PW] M. A. Picardello, W. Woess, Martin boundaries and random walks: ends of trees and groups, Trans. Amer. Math. Soc. 302 (1987), 185{205. [Re] D. Revuz, Markov Chains, 2nd revised ed., North-Holland, Amsterdam, 1984. [Su1] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225-255. [Su2] D. Sullivan, On the ergodic theory at in nity of an arbitrary discrete group of hyperbolic motions, Ann. Math. Studies 97 (1980), 465{496. [Su3] D. Sullivan, The density at in nity of a discrete group of hyperbolic motions, Publ. Math. IHES 50 (1979), 171{202. [Y] C. B. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, preprint, 1992. Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, Great Britain