Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry
aa r X i v : . [ m a t h . M G ] J un HOPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OFHADAMARD SPACES WITH A RANK ONE ISOMETRY
Gabriele Link ∗ Institut f¨ur Algebra und GeometrieKarlsruhe Institute of Technology (KIT)Englerstr. 2, 76 131 Karlsruhe, Germany
Abstract.
Let X be a proper Hadamard space and Γ < Is( X ) a non-elementary discrete group of isometries with a rank one isometry. We discussand prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set ofparametrized geodesics of the quotient Γ (cid:15) X and with respect to Ricks’ measureintroduced in [35]. This generalizes previous work of the author and J. C. Pi-caud on Hopf-Tsuji-Sullivan dichotomy in the analogous manifold setting andwith respect to Knieper’s measure. Introduction
Let (
X, d ) be a proper Hadamard space and Γ < Is( X ) a discrete group. Let G denote the set of parametrized geodesic lines in X endowed with the compact-open topology (which can be identified with the unit tangent bundle SX if X is aRiemannian manifold) and consider the action of R on G by reparametrization. Thisaction induces a flow g Γ on the quotient space Γ (cid:15) G . Let m Γ be an appropriate Radonmeasure on Γ (cid:15) G which is invariant by the flow g Γ . Hopf-Tsuji-Sullivan dichotomythen states that – under certain conditions on the space X and the group Γ –there are precisely two mutually exclusive possibilities for the dynamical system(Γ (cid:15) G , g Γ , m Γ ): Either it is conservative (that is almost every orbit is recurrent) andergodic (which means that the only invariant sets have zero or full measure) or it isdissipative (that is almost every orbit is divergent) and non-ergodic. For a precisedefinition of the previous notions the reader is referred to Section 5.The story of Hopf-Tsuji-Sullivan dichotomy probably began with Poincar´e’s re-currence theorem applied to Riemann surfaces and with Hopf’s seminal work laterin the 1930’s (see [19] and [20]). For quotients of the hyperbolic plane by Fuchsiangroups it was observed that with respect to Liouville measure the geodesic flowis either conservative and ergodic or dissipative and non-ergodic. Later, with theinvention of the remarkable Patterson-Sullivan measures on the boundary of X (see[32] and [41] for the original constructions, then [44], [21], [16] for extensions, and[36] for a clear account and deep applications of this theory) and then the construc-tion of Bowen-Margulis measure on Γ (cid:15) SX using these, generalizations to a widerclass of spaces and groups have been obtained by several authors. Among them Ionly want to mention here the work of M. Coornaert and A. Papadopoulos ([16])which deals with locally compact metric trees and the work of V. Kaimanovich([21]) in the setting of Gromov hyperbolic spaces with some additional properties; Mathematics Subject Classification.
Primary: 37D40, 20F67; Secondary: 37D25.
Key words and phrases. rank one space, Hopf-Tsuji-Sullivan dichotomy, geodesic currents,Patterson-Sullivan measures. these were probably the first ones considering non-Riemannian spaces. T. Roblin([36, Th´eor`eme 1.7]) then gave a unified version for all proper CAT( − manifolds by discrete isometry groups containing an element which trans-lates a geodesic without parallel perpendicular Jacobi field and with respect toKnieper’s measure ([23]) on the unit tangent bundle. The main goal of the presentpaper is to prove Hopf-Tsuji-Sullivan dichotomy in the setting of proper Hadamard spaces with a rank one isometry (that is an isometry translating a geodesic whichdoes not bound a flat half-plane) and hence to generalize the Main Theorem of[29] to non-Riemannian spaces; compared to [29] we also impose an a priori weakercondition on the discrete group Γ of the Hadamard manifold X : In fact, we onlyneed a discrete group with infinite limit set which contains the fixed point of a rankone isometry of X . So in particular X need not a priori possess a geodesic withoutparallel perpendicular Jacobi field, but only one without a flat half-plane. However,this can only happen when X does not admit a quotient of finite volume accordingto the rank rigidity theorem of Ballmann [4] and Burns-Spatzier [14], which assertsthat otherwise X has a geodesic without parallel perpendicular Jacobi field.Even though some of the results from the above mentioned paper [29] remain truein this more general setting, there are several obstructions occurring when singularspaces are involved. The probably most important one is the fact that Knieper’smeasure cannot be constructed without a volume form on the closed and convexsubsets corresponding to the parallel sets of geodesic lines. We will therefore followhere the construction proposed by R. Ricks in [35] and first define weak Bowen-Margulis measure on the quotient Γ (cid:15) [ G ] of parallel classes of parametrized geodesiclines by Γ. With respect to this measure we have the following Theorem A.
Let X be a proper Hadamard space and Γ < Is ( X ) a discrete groupwith the fixed point of a rank one isometry of X in its infinite limit set. Then withrespect to Ricks’ weak Bowen-Margulis measure either the geodesic flow on Γ (cid:15) [ G ] isconservative, or it is dissipative and non-ergodic unless the measure is supported ona single orbit by the geodesic flow on Γ (cid:15) [ G ] . Notice that since Ricks’ construction of weak Bowen-Margulis measure dependson the choice of a conformal density, a priori there may exist many distinct weakBowen-Margulis measures. In the conservative case however, it is well-known thatup to scaling there exists only one conformal density; hence there is precisely oneRicks’ weak Bowen-Margulis measure in this setting.We remark that we do not manage to deduce ergodicity from conservativity inthis weakest setting (only requiring the fixed point of an arbitrary rank one isometryin the limit set of Γ) as neither the Hopf argument nor Kaimanovich’s method forthe proof of Theorem 2.5 in [21] can be applied in this case. However, if X is geodesically complete then thanks to Proposition 1 this weak assumption impliesthe existence of a zero width rank one geodesic (that is one which does not evenbound a flat strip ) with extremities in the limit set of Γ. Under this additionalassumption any weak Bowen-Margulis measure induces a so-called Ricks’ Bowen-Margulis measure m Γ on the quotient Γ (cid:15) G . Notice that by the remark followingTheorem A there is only one Ricks’ Bowen-Margulis measure in the conservativecase. We finally get Theorem 10.2, the full Hopf-Tsuji-Sullivan dichotomy includingergodicity in the conservative case; a short version reads as follows: OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 3
Theorem B.
Let X be a proper Hadamard space and Γ < Is ( X ) a discrete groupwith the fixed point of a rank one isometry of X and the extremities of a zero width rank one geodesic in its infinite limit set. Then with respect to any Ricks’ Bowen-Margulis measure either the geodesic flow on Γ (cid:15) G is conservative and ergodic, or itis dissipative and non-ergodic unless the measure is supported on a single orbit bythe geodesic flow in Γ (cid:15) G . We finally want to mention here that if X is a Hadamard manifold , then in theconservative case Ricks’ Bowen-Margulis measure m Γ is equal to Knieper’s measurewhich was used in [29]. If moreover Γ is cocompact, then Knieper’s work [23] impliesthat the Rick’s Bowen-Margulis measure is the unique measure of maximal entropyon the unit tangent bundle Γ (cid:15) G .We summarize now what is known (from the Main Theorem of [29] and Theo-rem B above) in the special case of Hadamard manifolds : Theorem C.
Let X be a Hadamard manifold and Γ < Is ( X ) a discrete group withthe fixed point of an arbitrary rank one isometry of X in its infinite limit set. Theneither Knieper’s measure and Ricks’ Bowen-Margulis measure on Γ (cid:15) G coincide, andthe geodesic flow is conservative and ergodic with respect to this measure, or thegeodesic flow is dissipative with respect to any Knieper’s measure and with respectto any Ricks’ Bowen-Margulis measure on Γ (cid:15) G . Moreover, in the second case itis non-ergodic unless the considered measure is supported on a single orbit by thegeodesic flow. Again, in the dissipative case there may be several choices for Knieper’s measureand for Ricks’ Bowen-Margulis measure on Γ (cid:15) G as both measures are constructedfrom a conformal density. And even if the same conformal density is used in theconstruction, Knieper’s measure and Ricks’ Bowen-Margulis measure might be dif-ferent.Actually, in this article we will consider slightly more general classes of measureson Γ (cid:15) [ G ] respectively Γ (cid:15) G : Instead of using the geodesic current associated toa conformal density for the construction we allow for an arbitrary quasi-productgeodesic current (see Section 5 for a precise definition).The paper is organized as follows: In Section 2 we fix some notation and recallbasic facts concerning Hadamard spaces; in Section 3 the notion of rank one isometryis recalled and basic properties are listed.Section 4 discusses conditions under which a subgroup Γ of the isometry groupof a proper Hadamard space X is rank one (that is contains a pair of independentrank one elements), and under which hypotheses the presence of a rank one geodesicof zero width in X with extremities in the limit set of Γ can be guaranteed. Thissection is of independent interest.In Section 5 basic notions and useful facts from ergodic theory and dynamicalsystems are recalled, and the important notion of quasi-product geodesic currentis introduced. We also recall from [35] Ricks’ construction of a geodesic flow in-variant measure associated to such a geodesic current first on the quotient Γ (cid:15) [ G ]of parallel classes of parametrized geodesic lines and finally on the quotient Γ (cid:15) G ofparametrized geodesic lines. Section 6 deals with the relation between the radiallimit set of the group Γ and recurrence in Γ (cid:15) [ G ] respectively Γ (cid:15) G . We deduce thecrucial Theorem 6.7, which in particular implies that for a rank one group Γ withthe extremity of a zero width rank one geodesic in its limit set any conservativequasi-product geodesic current µ is supported on the set of end point pairs of zero GABRIELE LINK width rank one geodesics. In Section 7 we use the Hopf argument to show thatunder the presence of a zero width rank one geodesic with extremities in the limitset conservativity of a quasi-product geodesic current µ satisfying a mild growthcondition implies ergodicity of the geodesic flow with respect to the associated geo-desic flow invarant Ricks’ measure. Compared to the classical case a few technicalissues need to be addressed there.In Section 8 we then specialize to geodesic currents coming from a conformaldensity. We recall a few properties of conformal densities and prove Proposition 5,which states that for convergent groups every Ricks’ measure on Γ (cid:15) [ G ] is dissipa-tive. Section 9 is devoted to the proof of Proposition 7, namely that divergentgroups always induce conservative Ricks’ measure. The minimal requirement thatΓ contains only a rank one element of arbitrary width makes the proof a bit moretechnical than it would be with the presence of a zero width geodesic with extremi-ties in the limit set; however, it is needed in this form to obtain Theorem 10.1 whichis Theorem A above. In the final section 10 we summarize our results to deduceTheorems A, B and C. Following an idea of F. Dal’bo, M. Peign´e and J.P. Otal([17], [33]) we also show how to construct plenty of convergent discrete rank oneisometry groups of any Hadamard space admitting a rank one isometry.2. Preliminaries on Hadamard spaces
The purpose of this section is to introduce terminology and notation and tosummarize basic results about Hadamard spaces. Most of the material can befound in [11] and [5] (see also [8] in the special case of Hadamard manifolds and[35] for more recent results).Let (
X, d ) be a metric space. For y ∈ X and r > B y ( r ) ⊆ X the open ball of radius r centered at y ∈ X . A geodesic is a map σ from a closedinterval I ⊆ R or I = R to X such that d ( σ ( t ) , σ ( t ′ )) = | t − t ′ | for all t, t ′ ∈ I . Formore precision we use the term geodesic ray if I = [0 , ∞ ) and geodesic line if I = R .We will deal here with Hadamard spaces ( X, d ), that is complete metric spacesin which for any two points x, y ∈ X there exists a geodesic σ : [0 , d ( x, y )] → X with σ (0) = x and σ ( d ( x, y )) = y and in which all geodesic triangles satisfy theCAT(0)-inequality. This implies in particular that X is simply connected and thatthe geodesic joining an arbitrary pair of points in X is unique. Notice howeverthat in the non-Riemannian setting completeness of X does not imply that everygeodesic can be extended to a geodesic line, so X need not be geodesically complete.The geometric boundary ∂X of X is the set of equivalence classes of asymptoticgeodesic rays endowed with the cone topology (see for example Chapter II in [5]).We remark that for all x ∈ X and all ξ ∈ ∂X there exists a unique geodesic ray σ x,ξ with origin x = σ x,ξ (0) representing ξ .From here on we will require that X is proper; in this case the geometric boundary ∂X is compact and the space X is a dense and open subset of the compact space X := X ∪ ∂X . Moreover, the action of the isometry group Is( X ) on X naturallyextends to an action by homeomorphisms on the geometric boundary.If x, y ∈ X , ξ ∈ ∂X and σ is a geodesic ray in the class of ξ , we set(1) B ξ ( x, y ) := lim s →∞ (cid:0) d ( x, σ ( s )) − d ( y, σ ( s )) (cid:1) . This number exists, is independent of the chosen ray σ , and the function B ξ ( · , y ) : X → R , x
7→ B ξ ( x, y ) OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 5 is called the
Busemann function centered at ξ based at y (see also Chapter II in [5]).Obviously we have B g · ξ ( g · x, g · y ) = B ξ ( x, y ) for all x, y ∈ X and g ∈ Is( X ) , and the cocycle identity(2) B ξ ( x, z ) = B ξ ( x, y ) + B ξ ( y, z )holds for all x, y, z ∈ X .Since X is non-Riemannian in general, we consider (as a substitute of the unittangent bundle SX ) the set of parametrized geodesic lines in X which we will denote G . We endow this set with the metric d given by(3) d ( u, v ) := sup { e −| t | d (cid:0) v ( t ) , u ( t ) (cid:1) : t ∈ R } for u, v ∈ G ;this metric induces the compact-open topology, and every isometry of X naturallyextends to an isometry of the metric space ( G , d ).Moreover, there is a natural map p : G → X defined as follows: To a geodesicline v : R → X in G we assign its origin pv := v (0) ∈ X . Notice that p is proper,1-Lipschitz and Is( X )-equivariant; if X is geodesically complete, then p is surjective.For a geodesic line v ∈ G we denote its extremities v − := v ( −∞ ) ∈ ∂X and v + := v (+ ∞ ) ∈ ∂X the negative and positive end point of v ; in particular, we candefine the end point map ∂ : G → ∂X × ∂X, v ( v − , v + ) . We say that a point ξ ∈ ∂X can be joined to η ∈ ∂X by a geodesic v ∈ G if v − = ξ and v + = η . Obviously the set of pairs ( ξ, η ) ∈ ∂X × ∂X such that ξ and η can be joined by a geodesic coincides with ∂ G , the image of G under theend point map ∂ . It is well-known that if X is CAT( − ξ, η ) belongs to ∂ G and the geodesic joining ξ to η is unique upto reparametrization. In general however, the set ∂ G is much smaller compared to ∂X × ∂X minus the diagonal due to the possible existence of flat subspaces in X .For ( ξ, η ) ∈ ∂ G we denote by(4) ( ξη ) := p (cid:0) { v ∈ G : v − = ξ, v + = η } (cid:1) = p ◦ ∂ − ( ξ, η )the subset of points in X which lie on a geodesic line joining ξ to η . It is well-knownthat ( ξη ) = ( ηξ ) ⊆ X is a closed and convex subset of X which is isometric to aproduct C ( ξη ) × R , where C ( ξη ) = C ( ηξ ) is again a closed and convex set.In order to describe the sets ( ξη ) and C ( ξη ) more precisely and for later use weintroduce as in [35, Definition 5.4] for x ∈ X the so-called Hopf parametrization map(5) H x : G → ∂ G × R , v (cid:0) v − , v + , B v − ( x, v (0)) (cid:1) of G with respect to x . It is immediate that for a CAT( − X this map isa homeomorphism; in general it is only continuous and surjective. Moreover, itdepends on the point x ∈ X as follows: If y ∈ X , v ∈ G and H x ( v ) = ( ξ, η, s ), then H y ( v ) = (cid:0) ξ, η, s + B ξ ( y, x ) (cid:1) by the cocycle identity (2) for the Busemann function (compare also [16, Section 3]).The Hopf parametrization map allows to define an equivalence relation ∼ on G as follows: If u, v ∈ G , then u ∼ v if and only if H o ( u ) = H o ( v ). Noticethat this definition does not depend on the choice of o ∈ X and that every point GABRIELE LINK ( ξ, η, s ) ∈ ∂ G × R uniquely determines an equivalence class [ v ] with v ∈ G . Moreover,the closed and convex set C ( ξη ) from above can be identified with the set(6) C v := p (cid:0) { u ∈ G : u ∼ v } (cid:1) ⊆ X, which we will call the transversal of v . We remark that for all w ∈ ∂ − ( ξ, η ) thetransversal C w is isometric to C v . Moreover, if X is CAT( −
1) then for all v ∈ G the transversal C v is simply a point; in general, the transversals can be unbounded.As stated in [35, Proposition 5.10] the Is( X )-action on G descends to an actionon ∂ G × R = H o ( G ) by homeomorphisms via γ ( ξ, η, s ) := (cid:0) γξ, γη, s + B γξ ( o, γo ) (cid:1) . Moreover, the action of Is( X ) is well-defined on the set of equivalence classes [ G ]of elements in G , and the (well-defined) map(7) [ G ] → ∂ G × R , [ v ] H o ( v )is an Is( X )-equivariant homeomorphism. For convenience we will frequently identify ∂ G × R with [ G ]. We also remark that the end point map ∂ : G → ∂X × ∂X inducesa well-defined map [ G ] → ∂X × ∂X which we will also denote ∂ .As in Definition 5.4 of [35] we will say that a sequence ( v n ) ⊆ G converges weakly to v ∈ G if and only if(8) v − n → v − , v + n → v + and B v − n (cid:0) o, v n (0) (cid:1) → B v − (cid:0) o, v (0) (cid:1) . Obviously, weak convergence v n → v is equivalent to the convergence [ v n ] → [ v ] in[ G ], and v n → v in G always implies [ v n ] → [ v ] in [ G ].The topological space G can be endowed with the geodesic flow ( g t ) t ∈ R which isnaturally defined by reparametrization of v ∈ G . In particular we have( g t v )(0) = v ( t ) for all v ∈ G and all t ∈ R . The geodesic flow induces a flow on the set of equivalence classes [ G ] which we willalso denote ( g t ) t ∈ R ; via the Is( X )-equivariant homeomorphism [ G ] → ∂ G × R theaction of the geodesic flow ( g t ) t ∈ R on [ G ] is equivalent to the translation action onthe last factor of ∂ G × R given by g t ( ξ, η, s ) := ( ξ, η, s + t ) . Facts about rank one isometries
The purpose of this section is to introduce the notion of rank one geodesic andrank one isometry. Many useful well-known facts about Hadamard spaces with arank one isometry are recalled. Most of the material can be found in [5] and [6](see also [3] for the special case of Hadamard manifolds and [35] for more recentresults).As in the previous section we assume that (
X, d ) is a proper Hadamard space. Ageodesic line v ∈ G is called rank one if its transversal C v is bounded. In this casethe number width( v ) := sup { d ( x, y ) : x, y ∈ C v } is called the width of v ; if C v reduces to a point, then v is said to have zero width.In the sequel we will use as in [35] the notation R := { v ∈ G : v is rank one } respectively Z := { v ∈ G : v is rank one of zero width } . OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 7
We remark that the existence of a rank one geodesic imposes severe restrictionson the Hadamard space X . For example, X can neither be a symmetric space orEuclidean building of higher rank nor a product of Hadamard spaces.Notice that if X is a Hadamard manifold , then there is a more restrictive notionof rank one: If v ∈ G the number J -rank( v ) is defined as the dimension of the vectorspace of parallel Jacobi fields along v (compare Section IV.4 in [5]); clearly, for all w in a sufficiently small neighborhood of v we have J -rank( w ) ≤ J -rank( v ). As in[29] we will call v ∈ G strong rank one if J -rank( v ) = 1, that is if v does not admita parallel perpendicular Jacobi field; we further define J := { v ∈ G : v is strong rank one } which is obviously a subset of Z . Notice that in general J 6 = Z : Take for examplea surface with negative Gaußian curvature except along a simple closed geodesicwhere the curvature vanishes; then the lift of the closed geodesic has zero width,but possesses a parallel perpendicular Jacobi field.The following important lemma states that even though we cannot join any twodistinct points in the geometric boundary ∂X of the Hadamard space X , givena rank one geodesic we can at least join all points in a neighborhood of its endpoints. More precisely, we have the following result which is a reformulation ofLemma III.3.1 in [5]: Lemma 3.1 (Ballmann) . Let v ∈ R be a rank one geodesic and c > width( v ) .Then there exist open disjoint neighborhoods U − of v − and U + of v + in X with thefollowing properties: If ξ ∈ U − and η ∈ U + then there exists a rank one geodesicjoining ξ and η . For any such geodesic w ∈ R we have d ( w ( t ) , v (0)) < c for some t ∈ R and width( w ) ≤ c . This lemma implies that the set R is open in G ; we emphasize that Z in generalneed not be an open subset of G : In every open neighborhood of a zero width rankone geodesic there may exist a rank one geodesic of arbitrarily small but strictlypositive width. However, if X is a Hadamard manifold , then J ⊆ Z is open in G (as the J -rank cannot be bigger in a suffiently small open neighborhood). SoLemma 3.1 has the following Corollary 1.
Let v ∈ J . Then there exist disjoint neighborhoods U − of v − and U + of v + in X such that any pair of points ( ξ, η ) ∈ U − × U + can be joined by ageodesic u ∈ J . We will also need the following result due to R. Ricks; recall that ( v n ) → v weakly as defined in (8) means that [ v n ] → [ v ] in [ G ]. Lemma 3.2 ([35], Lemma 5.9) . If a sequence ( v n ) ⊆ G converges weakly to v ∈ R ,then some subsequence of ( v n ) converges to some u ∼ v . Notice that this lemma implies that the restriction of the Hopf parametrizationmap (5) to the subset R is closed, hence a topological quotient map.In combination with Lemma 8.4 in [35] we get the following statement concerningtransversals of a weakly convergent sequence in G : Lemma 3.3.
If a sequence ( v n ) ⊆ G converges weakly to v ∈ R , then some subse-quence of ( C v n ) converges, in the Hausdorff metric, to a closed subset A ⊆ C v . From this we immediately get the following complement to Lemma 3.1:
GABRIELE LINK
Lemma 3.4.
Let v ∈ Z and (cid:0) ( ξ n , η n ) (cid:1) ⊆ ∂X × ∂X be a sequence converging to ( v − , v + ) . Then for n sufficiently large ( ξ n , η n ) ∈ ∂ R and some subsequence of (cid:0) C ( ξ n η n ) (cid:1) converges, in the Hausdorff metric, to a point. Definition 3.5.
An isometry γ = id of X is called axial if there exists a constant ℓ = ℓ ( γ ) > v ∈ G such that γv = g ℓ v . We call ℓ ( γ ) the translationlength of γ , and v an invariant geodesic of γ . The boundary point γ + := v + (whichis independent of the chosen invariant geodesic v ) is called the attractive fixed point ,and γ − := v − the repulsive fixed point of γ .An axial isometry h is called rank one if one (and hence any) invariant geodesic of h belongs to R ; the width of h is then defined as the width of an arbitrary invariantgeodesic of h . h is said to have zero width if up to reparametrization h has onlyone invariant geodesic.Notice that if γ ∈ Is( X ) is axial, then ∂ − ( γ − , γ + ) ⊆ G is the set of parametrizedinvariant geodesics of γ , and every axial isometry e γ commuting with γ satisfies p∂ − ( e γ − , e γ + ) = p∂ − ( γ − , γ + ). If h is rank one, then the fixed point set of h equals { h − , h + } , and every axial isometry commuting with h belongs to the subgroup h h i < Is( X ) generated by h .The following important lemma describes the north-south dynamics of rank oneisometries: Lemma 3.6. ( [5] , Lemma III.3.3) Let h be a rank one isometry. Then (a) every point ξ ∈ ∂X \ { h + } can be joined to h + by a geodesic, and all thesegeodesics are rank one, (b) given neighborhoods U − of h − and U + of h + in X there exists N ∈ N suchthat h − n ( X \ U + ) ⊆ U − and h n ( X \ U − ) ⊆ U + for all n ≥ N . The following lemma shows that under the presence of a rank one geodesic in X with Is ( X ) -dual end points (the interested reader is referred to Section III.1 in [5]for a definition) the rank one isometries are numerous: Lemma 3.7. ( [5] , Lemma III.3.2) Let v ∈ R be a rank one geodesic, and ( g n ) ⊆ Is ( X ) a sequence of isometries such that g n x → v + and g − n x → v − for one (andhence any) x ∈ X . Then, for n sufficiently large, g n is rank one with an invariantgeodesic v n such that v + n → v + and v − n → v − . We next prepare for an extension of Lemma 3.6 (a) which replaces the fixed point h + of the rank one isometry h by the end point of a certain geodesic: Definition 3.8 (compare Section 5 in [35]) . Let
G <
Is( X ) be any subgroup. Anelement v ∈ G is said to (weakly) G -accumulate on u ∈ G if there exist sequences( g n ) ⊆ G and ( t n ) ր ∞ such that g n g t n v converges (weakly) to u as n → ∞ ; v issaid to be (weakly) G -recurrent if v (weakly) G -accumulates on v .Notice that if v is an invariant geodesic of an axial isometry γ ∈ Is( X ), then v is h γ i -recurrent and hence in particular Is( X )-recurrent. Moreover, if v ∈ G weakly G -accumulates on u ∈ R , then by Lemma 3.2 v G -accumulates on some element w ∼ u . However, in general v ∈ G weakly G -recurrent does not imply that somerepresentative of the equivalence class [ v ] is G -recurrent. Even in the case v ∈ R itis possible that every representative u of the class [ v ] G -accumulates on w ∼ u with w = u .The following statements show the relevance of the previous notions. OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 9
Lemma 3.9 ([35], Lemma 6.10) . If w ∈ G Is ( X ) -accumulates on v ∈ G , then thereexists an isometric embedding C w ֒ → C v which maps w (0) to v (0) . Notice that if v ∈ R is weakly G -recurrent for some subgroup G <
Is( X ), thenevery w ∈ G with w + = v + G -accumulates on an element u ∼ v according toLemma 6.9 in [35]. Hence we have Lemma 3.10 (Corollary 6.11 in [35]) . If v ∈ R is weakly Is ( X ) -recurrent, then forevery w ∈ G with w + = v + there exists an isometric embedding C w ֒ → C v . Moreover, the proof of Lemma 6.12 in [35] shows that every point ξ ∈ ∂X \ { v + } can be joined to v + by a geodesic w ∈ G . So we finally get Lemma 3.11. If v ∈ R is weakly Is ( X ) -recurrent then for every ξ ∈ ∂X \ { v + } there exists w ∈ R with width( w ) ≤ width( v ) such that w − = ξ and w + = v + . Rank one groups
Let X be a proper Hadamard space and Γ < Is( X ) an arbitrary subgroup. The geometric limit set L Γ of Γ is defined by L Γ := Γ · x ∩ ∂X, where x ∈ X is anarbitrary point.If X is a CAT( − < Is( X ) is called non-elementary if itslimit set is infinite and if Γ does not globally fix a point in L Γ . It is well-knownthat this implies that Γ contains two axial isometries with disjoint fixed point sets(which are actually rank one of zero width as G = Z for any CAT( − Definition 4.1.
We say that two rank one isometries g, h ∈ Is( X ) are independent if and only if { g + , g − } ∩ { h + , h − } 6 = ∅ (see for example Section 2 of [27]).Moreover, a group Γ < Is( X ) is called rank one if Γ contains a pair of independentrank one elements.Obviously, if X is CAT( −
1) then every non-elementary isometry group is rankone. In general however, the notion of rank one group seems very restrictive at firstsight. The goal of this section – which may be of independent interest – is to discussconditions which ensure that Γ is a rank one group.
Lemma 4.2.
Let Γ < Is ( X ) be an arbitrary subgroup. If L Γ contains the positiveend point v + of a weakly Is ( X ) -recurrent element v ∈ R , and if v + is not globallyfixed by Γ , then Γ contains a rank one isometry.Proof. Let v ∈ R be weakly Is( X )-recurrent and x ∈ X . As v + ∈ L Γ there existsa sequence ( γ n ) ⊆ Γ such that γ n x → v + as n → ∞ . Passing to a subsequenceif necessary we may assume that γ − n x converges, say to a point ξ ∈ X whichobviously belongs to L Γ ⊆ ∂X . If ξ = v + , there exists γ ∈ Γ such that γξ = v + since Γ does not globally fix v + . Replacing the sequence ( γ n ) by ( γ n γ − ) in thiscase we may assume that ξ = v + . According to Lemma 3.11 there exists w ∈ R such that w − = ξ and w + = v + . Lemma 3.7 then states that for n sufficientlylarge γ n is rank one with an invariant geodesic v n such that v + n → w + = v + and v − n → w − = ξ as n → ∞ . Since the geodesic w is rank one, the geodesics v n arerank one for n sufficiently large by Lemma 3.1. This implies that for some fixed n large enough the element γ n ∈ Γ is rank one. (cid:3)
Notice that the conclusion is obviously true when v + is a fixed point of a rankone isometry of X . The following statements show that a group is rank one undervery weak conditions. Lemma 4.3. If Γ < Is ( X ) neither globally fixes a point in ∂X nor stabilizes ageodesic line in X , and if L Γ contains the positive end point v + of a weakly Is ( X ) -recurrent element v ∈ R , then Γ contains a pair of independent rank one elements.Proof. Since X is proper and Γ < Is( X ) contains a rank one element by the pre-vious lemma, Proposition 3.4 of [15] applies: Its first possibility is excluded by theassumption that Γ neither globally fixes a point in ∂X nor stabilizes a geodesic linein X , hence Γ contains a pair of independent rank one elements. (cid:3) Lemma 4.4. A discrete subgroup Γ < Is ( X ) is rank one if and only if its limit set L Γ is infinite and contains the positive end point v + of a weakly Is ( X ) -recurrentelement v ∈ R .Proof. We first assume that L Γ is infinite and contains the positive end point v + of a weakly Is( X )-recurrent element v ∈ R . As Γ is discrete and L Γ is infinite, Γcannot globally fix a point in ∂X nor stabilize a geodesic line in X , so Lemma 4.3above implies that Γ is rank one. The other direction is obvious. (cid:3) The proof of the following criterion relies heavily on the work of R. Ricks:
Proposition 1. If X is geodesically complete and Γ < Is ( X ) is a discrete rank onegroup, then Z Γ := { v ∈ Z : v − , v + ∈ L Γ } 6 = ∅ . Proof.
We first notice that the proof of Theorem III.2.3 in [5] shows that the geodesicflow restricted to G Γ := { v ∈ G : v − , v + ∈ L Γ } is topologically transitive mod Γ; this means that there exists v ∈ G Γ such that forany u ∈ G Γ v Γ-accumulates on u .We first claim that the element v ∈ G Γ as above belongs to R : We choose arank one element h ∈ Γ and an invariant geodesic u ∈ G Γ of h and neighborhoods U − , U + ⊆ X of h − , h + as in Lemma 3.1. In particular, every w ∈ G with ( w − , w + ) ∈ U − × U + satisfies w ∈ R . As v Γ-accumulates on u there exist sequences ( γ n ) ⊆ Γ,( t n ) ր ∞ such that γ n g t n v → u and hence in particular γ n ( v − , v + ) → ( u − , u + ) =( h − , h + ) as n → ∞ . This implies γ n ( v − , v + ) ∈ U − × U + ⊆ ∂ R for some n sufficiently large and therefore v ∈ R .Assume for a contradiction that v / ∈ Z ; then there exists v ∼ v with v = v . Wewill further denote v C ∈ p − C v = { w ∈ R : w ∼ v } the central geodesic definedby the condition that its origin v C (0) is the unique circumcenter of the boundedclosed and convex set C v ⊆ X (compare also Section 5 in [35]). As v C , v ∈ G Γ , v Γ-accumulates both on v C and on v ; so according to Lemma 3.9 there exist isometricembeddings ι : C v ֒ → C v C , ι : C v ֒ → C v with ι (cid:0) v (0) (cid:1) = v C (0) and ι (cid:0) v (0) (cid:1) = v (0). Since C v C = C v = C v , the maps ι and ι are surjective by Theorem 1.6.15 in [12] and hence isometries. As the circumcenterof C v is invariant by isometries of C v we first get v (0) = ι − (cid:0) v C (0) (cid:1) = v C (0) , which implies v (0) = ι (cid:0) v (0) (cid:1) = ι (cid:0) v C (0) (cid:1) = v C (0) = v (0) . This is a contradiction to the choice of v = v , so we conclude that v ∈ Z . (cid:3) OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 11
Notice that a discrete rank one group Γ with Z Γ = ∅ need not possess a zerowidth rank one isometry since Z is not open in G . However, as for a Hadamard manifold the set J of vectors not admitting a parallel perpendicular Jacobi field isopen in G , we have the following Lemma 4.5. If X is a manifold and Γ < Is ( X ) a discrete rank one group suchthat J Γ := { v ∈ J : v − , v + ∈ L Γ } 6 = ∅ , then Γ contains a pair of independent rank one elements with strong rank oneinvariant geodesics (which necessarily have zero width).Proof. Since X is geodesically complete, the geodesic flow restricted to G Γ := { v ∈ G : v − , v + ∈ L Γ } is topologically transitive mod Γ; this means that there exists v ∈ G Γ such that forany u ∈ G Γ v Γ-accumulates on u . Assume for a contradiction that v / ∈ J ; then γg t v / ∈ J for all γ ∈ Γ and for all t ∈ R . But since v Γ-accumulates on u ∈ J Γ thisimplies J -rank( u ) ≥ v ∈ J Γ .Since v − , v + ∈ L Γ , there exists a sequence ( γ n ) ⊆ Γ such that γ n x → v + and γ − n x → v − for some x ∈ X (see for example the proof of Proposition 3.5 in [15]).By Lemma 3.7, for n sufficiently large γ n is rank one with invariant geodesic v n suchthat ( v − n , v + n ) → ( v − , v + ) as n → ∞ . So according to Corollary 1 we have v n ∈ J for n sufficiently large, hence there exists a rank one element γ n with a strongrank one invariant geodesic. As Γ is rank one there exists one element (actuallyan infinite number) in Γ not commuting with γ n , and conjugating γ n by such anelement provides another rank one isometry in Γ independent from γ n which alsohas a strong rank one invariant geodesic. (cid:3) This implies that the hypothesis of the Main Theorem in [29] is satisfied forHadamard manifolds X with a rank one group Γ < Is( X ) such that J Γ = ∅ ; we willsee later that the conclusion of the Main Theorem in [29] remains true under theweaker condition that Γ < Is( X ) is an arbitrary rank one group.5. Basic notions in ergodic theory and geodesic currents
In this section we want to recall a few general notions from topological dynamicsand ergodic theory which will be needed later; our main references here are [19] and[21].Let Ω be a locally compact and σ -compact Hausdorff topological space and ϕ a flow on Ω, that is a continuous map ϕ : R × Ω → Ω such that ϕ (0 , ω ) = ω and ϕ (cid:0) s, ϕ ( t, ω ) (cid:1) = ϕ ( s + t, ω ) for all s, t ∈ R and all ω ∈ Ω.A point ω ∈ Ω is said to be positively recurrent respectively negatively recurrent if there exists a sequence ( t n ) ր ∞ of real numbers such that ϕ t n ω = ϕ ( t n , ω ) → ω respectively ϕ − t n ω = ϕ ( − t n , ω ) → ω ; ω ∈ Ω is said to be positively divergent respectively negatively divergent if for everycompact set K ⊆ Ω there exists a constant
T > t ≥ Tϕ t ω = ϕ ( t, ω ) / ∈ K respectively ϕ − t ω = ϕ ( − t, ω ) / ∈ K. Assume that M is a Borel measure on Ω invariant by the flow ϕ . Then the Hopfdecomposition theorem (see for instance [25, Theorem 3.2],[19, Satz 13.1] ) asserts that the space Ω decomposes into a disjoint union of ϕ -invariant Borel sets Ω C andΩ D which satisfy the following properties:(C) There does not exist a Borel subset E ⊆ Ω C with M ( E ) > (cid:0) ϕ k ( E ) (cid:1) k ∈ Z are pairwise disjoint.(D) There exists a Borel set W ⊆ Ω D such that Ω D is the disjoint union of sets( W k ) k ∈ Z , where each W k is a translate of W under the flow ϕ .According to Poincar´e’s recurrence theorem (see for example [19, Satz 13.2]) M -almost every point of Ω C is positively recurrent. On the other hand, by Hopf’sdivergence theorem (see again [19, Satz 13.2]), M -almost every point of Ω D is posi-tively divergent. This implies in particular that the sets Ω C and Ω D are unique upto sets of measure zero.The dynamical system (Ω , ϕ, M ) is said to be conservative if M (Ω D ) = 0, and dissipative if M (Ω C ) = 0. Notice that if the measure M is finite, then due to (D)above (Ω , ϕ, M ) is conservative. Moreover, since the decomposition is the samefor ϕ and for ϕ − , Poincar´e’s recurrence theorem and Hopf’s divergence theoremimply that M -almost every point of Ω C is positively and negatively recurrent, and M -almost every point of Ω D is positively and negatively divergent. Moreover, if ρ ∈ L ( M ) is M -almost everywhere strictly positive, then – up to a set of measurezero – the conservative part Ω C can be written asΩ C = { ω ∈ Ω : Z ∞ ρ ( ϕ t ω )d t = ∞} . Finally, the dynamical system (Ω , ϕ, M ) is called ergodic if every ϕ -invariantBorel set E ⊆ Ω either satisfies M ( E ) = 0 or M (Ω \ E ) = 0. Hence if a dynamicalsystem (Ω , ϕ, M ) is ergodic, then it is either conservative or dissipative; the secondpossibility can only occur for an infinite measure M which is supported on a singleorbit { ϕ t ω : t ∈ R } with ω ∈ Ω . In Section 7 we will need the following generalization of the Birkhoff ergodictheorem which is stated and proved on p. 53 in [19]:
Theorem 5.1 (Hopf’s individual ergodic theorem) . Assume that (Ω , ϕ, M ) is con-servative, and let ρ ∈ L ( M ) be a function which is strictly positive M -almosteverywhere.Then for any function f ∈ L ( M ) the limits f ± ( ω ) = lim T → + ∞ R T f ( ϕ ± t ( ω ))d t R T ρ ( ϕ ± t ( ω ))d t exist and are equal for M -almost every ω ∈ Ω . Moreover, the functions f + , f − are measurable and flow invariant, ρ · f + , ρ · f − ∈ L ( M ) , and for every boundedmeasurable flow-invariant function h we have Z Ω ρ ( ω ) f ± ( ω ) h ( ω )d M ( ω ) = Z Ω f ( ω ) h ( ω )d M ( ω ) . Finally, (Ω , ϕ, M ) is ergodic if and only if for every function f ∈ L ( M ) the asso-ciated limit function f + is constant M -almost everywhere. We now want to recall the concept of geodesic current introduced for example in[21]. From here on we let X be a proper Hadamard space and Γ < Is( X ) a discrete OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 13 group. We will also use the notation introduced in Section 2 and Section 3. Thegeodesic flow on the quotient Γ (cid:15) G will be denoted g Γ = ( g t Γ ) t ∈ R .Recall that a Borel measure on a locally compact Hausdorff space is called Radon if it is finite for all compact subsets.
Definition 5.2 (compare Definitions 2.3 and 2.5 in [21]) . A geodesic current on Γ (cid:15) X is a Γ-invariant Radon measure on ∂ G ⊆ ∂X × ∂X .A geodesic current µ is said to be a quasi-product geodesic current , if there existprobability measures µ − , µ + on ∂X such that µ is absolutely continuous withrespect to the product measure µ − ⊗ µ + .A geodesic current µ hence yields a dynamical system ( ∂ G , Γ , µ ) which is closelyrelated to the dynamical system ( ∂X × ∂X, Γ , µ − ⊗ µ + ) with the diagonal actionof Γ on ∂X × ∂X . As in [36, p.17] a Borel set W ⊆ ∂ G is called wandering if for µ -almost every ( ξ, η ) ∈ W the number { γ ∈ Γ : γ ( ξ, η ) ∈ W } is finite . The Γ-action on ∂ G is called dissipative if up to sets of measure zero the set ∂ G is a countable union of wandering sets; it is called conservative if every wanderingsubset W ⊆ ∂ G satisfies µ ( W ) = 0.Let µ be a geodesic current such that for µ -almost every ( ξ, η ) ∈ ∂ G a geodesicflow invariant Radon measure λ ( ξη ) on the closed and convex subset ( ξη ) ⊆ X exists.Then we get a Γ-invariant and geodesic flow invariant Borel measure m on G byintegrating µ with respect to the measure λ ( ξη ) along the sets ( ξη ) ⊆ X , that is viathe assignment m ( E ) := Z ∂ G λ ( ξη ) (cid:0) p ( E ) ∩ ( ξη ) (cid:1) d µ ( ξ, η ) for any Borel set E ⊆ G . Notice that by continuity of the maps p : G → X and ∂ : G → ∂ G the Borel measure m is Radon as well. If ( ξ, η ) ∈ ∂ Z , then we use the convention that the Radonmeasure λ ( ξη ) on ( ξη ) ∼ = R is Lebesgue measure on R (which in addition is innerand outer regular).The Radon measure m then induces a geodesic flow invariant measure m Γ onthe quotient Γ (cid:15) G which we will call a Knieper’s measure on Γ (cid:15) G for the followingreason: In [23], G. Knieper constructed for a Hadamard manifold X a measure onΓ (cid:15) G precisely in this way with λ ( ξη ) the induced Riemannian volume element onthe submanifolds ( ξη ) ⊆ X and µ the quasi-product geodesic current induced by aconformal density for Γ (see Section 8 for the precise definition).Unfortunately, if X is not a manifold then in general there is no natural geodesicflow invariant measure on the closed and convex subsets ( ξη ) for ( ξ, η ) ∈ ∂ ( G \ Z ).Hence we will follow Ricks’ approach to obtain from a geodesic current a geodesicflow and Γ-invariant measure on the set of parallel classes of parametrized geodesiclines [ G ]: Given a geodesic current µ on ∂ G = ∂ [ G ] we want to define a Radonmeasure m on [ G ] ∼ = ∂ G × R by µ ⊗ λ , where λ denotes Lebesgue measure on R .However, the Γ-action on [ G ] need not be proper: If Γ contains an axial isometry γ with invariant geodesic w ∈ G \ R whose image w ( R ) belongs to an isometric copy E ⊂ ( γ − γ + ) of a Euclidean plane, then for any geodesic u ∈ G orthogonal to w andwith image u ( R ) ⊆ E we have γ k u ∼ u and hence γ k [ u ] = [ u ] for all k ∈ Z . So inparticular we do not necessarily obtain from m a geodesic flow invariant measureon the quotient Γ (cid:15) [ G ]. For that reason we will consider only geodesic currents µ which are defined on ∂ R instead of ∂ G . According to Lemma 3.1, Γ acts properly on [ R ] ∼ = ∂ R× R which admits a propermetric. Since the action is by homeomorphisms and preserves the Borel measure m = µ ⊗ λ , there is (see for instance, [34, Appendix A]) a unique Borel quotientmeasure m Γ on Γ (cid:15) [ R ] satisfying the characterizing property Z ¯ A e h d m = Z Γ (cid:15) [ R ] (cid:0) h · f ¯ A (cid:1) d m Γ for all Borel sets ¯ A ⊆ [ R ] and Γ-invariant Borel maps e h : [ R ] → [0 , ∞ ] and e f ¯ A : [ R ] → [0 , ∞ ] defined by e f ¯ A ([ v ]) := { γ ∈ Γ : γ [ v ] ∈ ¯ A } for [ v ] ∈ R , andwith h and f ¯ A the maps on Γ (cid:15) [ R ] induced from e h and e f ¯ A .According to the characterizing property above, a Borel set ¯ A ⊂ [ G ] satisfies m ( ¯ A ) = 0 if and only if its projection ¯ A Γ to Γ (cid:15) [ G ] satisfies m Γ ( ¯ A Γ ) = 0. So in factwe can consider m Γ as a Borel measure on Γ (cid:15) [ G ]; we will call m Γ the weak Ricks’measure associated to the geodesic current µ on ∂ R .Our final goal is to construct from a weak Ricks’ measure m Γ a geodesic flowinvariant measure on Γ (cid:15) G . So let us first remark that Z ⊆ R is a Borel subset bysemicontinuity (see Lemma 3.3) of the width function; as H o R : R → ∂ R× R ∼ = [ R ]is a topological quotient map by Lemma 3.2, [ Z ] ⊆ [ R ] is also a Borel subset.Notice also that H o Z : Z → ∂ Z × R ∼ = [ Z ] is a homeomorphism. So if Γ (cid:15) [ Z ] haspositive mass with respect to the weak Ricks’ measure m Γ we may define (as in [35,Definition 8.12]) a geodesic flow and Γ-invariant measure m on G by setting(9) m ( E ) := m (cid:0) H o ( E ∩ Z ) (cid:1) for any Borel set E ⊆ G ;this measure m then induces the Ricks’ measure m on Γ (cid:15) G .Notice that in general m Γ (Γ (cid:15) [ Z ]) = 0 is possible; obviously this is always thecase when Z = ∅ . However, we will see later that under certain conditions the Ricks’measure is actually equal to the weak Ricks’ measure used for its construction.6. The radial limit set and recurrence
As before X will always be a proper Hadamard space and Γ < Is( X ) a discreterank one group. We further fix a base point o ∈ X . We will begin this section witha few definitions.A point ξ ∈ ∂X is called a radial limit point if there exists c > γ n ) ⊆ Γ and ( t n ) ր ∞ such that(10) d (cid:0) γ n o, σ o,ξ ( t n ) (cid:1) ≤ c for all n ∈ N . Notice that by the triangle inequality this condition is independent of the choiceof o ∈ X . The radial limit set L radΓ ⊆ L Γ of Γ is defined as the set of radial limitpoints.Recall the notion of (weakly) Γ-recurrent elements from Definition 3.8. Moreover,an element v ∈ G is called Γ -divergent if for every compact set K ⊆ G there exists T > t ≥ T g t v / ∈ [ γ ∈ Γ γK ;it is called weakly Γ -divergent if for every compact set K ⊂ [ G ] there exists T > t ≥ T g t [ v ] / ∈ [ γ ∈ Γ γK. OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 15
For the convenience of the reader we state the following easy fact.
Lemma 6.1.
Let u ∈ G . Then u Γ -recurrent = ⇒ u + ∈ L radΓ = ⇒ u not Γ -divergent . We want to emphasize here that in general u weakly Γ-recurrent does not imply u + ∈ L radΓ , while u not Γ-divergent always implies u not weakly Γ-divergent. How-ever, if u ∈ R is weakly Γ-recurrent, then according to Lemma 3.2 u Γ-accumulatesto some w ∼ u . This again implies that w + = u + ∈ L radΓ and we get the following Lemma 6.2. If u ∈ R then u weakly Γ -recurrent = ⇒ u + ∈ L radΓ = ⇒ u not weakly Γ -divergent . In the sequel the following subsets of G will be convenient. Notice that for v ∈ G the reverse geodesic − v ∈ G is defined by − v ( s ) := v ( − s ) for all s ∈ R . G radΓ := { v ∈ G : v − ∈ L radΓ , v + ∈ L radΓ } , G recΓ := { v ∈ G : v and − v are Γ-recurrent } , G divΓ := { v ∈ G : v and − v are Γ-divergent } , G wrecΓ := { v ∈ G : v and − v are weakly Γ-recurrent } , G wdivΓ := { v ∈ G : v and − v are weakly Γ-divergent } . Notice that in general [ G wrecΓ ] ( [ G recΓ ] and even[ G wrecΓ ∩ R ] ( [ G recΓ ∩ R ]by the remark following Definition 3.8.From now on we will also deal with the quotient Γ (cid:15) G ; for the remainder of thissection we will therefore denote elements in the quotient by u, v, w and elements in G by e u, e v, e w . According to the definitions given in Section 5, v ∈ Γ (cid:15) G is positivelyand negatively recurrent if and only if every lift e v of v belongs to G recΓ ; v ∈ Γ (cid:15) G ispositively and negatively divergent if and only if every lift e v of v belongs to G divΓ .Similarly, [ v ] ∈ Γ (cid:15) [ G ] is positively and negatively recurrent if and only if for everylift [ e v ] ∈ [ G ] and every representative e u ∈ G of [ e v ] we have e u ∈ G wrecΓ ; [ v ] ∈ Γ (cid:15) [ G ] ispositively and negatively divergent if and only if for every lift [ e v ] ∈ [ G ] and everyrepresentative e u ∈ G of [ e v ] we have e u ∈ G wdivΓ .We now assume that m Γ is a Knieper’s measure on Γ (cid:15) G constructed from anarbitrary geodesic current µ and that m Γ is a weak Ricks’ measure on Γ (cid:15) [ G ] comingfrom a geodesic current µ defined on ∂ R . For the convenience of the reader westate and prove the following easy Lemma 6.3 (compare also Theorem 2.3 in [21]) . The dynamical systems (cid:0) Γ (cid:15) G , g Γ , m Γ (cid:1) respectively (cid:0) Γ (cid:15) [ G ] , g Γ , m Γ (cid:1) are (a) conservative if and only if µ (cid:0) ∂ ( G \ G radΓ ) (cid:1) = 0 , (b) dissipative if and only if µ ( ∂ G radΓ ) = 0 .Moreover, in the dissipative case the measures m Γ and m Γ are infinite, and thecorresponding dynamical systems are non-ergodic unless µ is supported on a singleorbit Γ · ( ξ, η ) ⊆ ∂ G .Proof. We first treat the dynamical system (cid:0) Γ (cid:15) G , g Γ , m Γ (cid:1) with Knieper’s measure m Γ ; let Ω D denote its dissipiative part and Ω C its conservative part. Then by Poincar´e’s recurrence theorem and Hopf’s divergence theorem we have m Γ (Ω D ) = m Γ (cid:0) Γ (cid:15) G divΓ (cid:1) and m Γ (Ω C ) = m Γ (cid:0) Γ (cid:15) G recΓ (cid:1) . Moreover, Lemma 6.1 implies G divΓ ⊆ G \ G radΓ and G recΓ ⊆ G radΓ , and as G = G radΓ ⊔ G \ G radΓ we get m Γ (Ω D ) = m Γ (cid:0) Γ (cid:15) ( G \ G radΓ ) (cid:1) and m Γ (Ω C ) = m Γ (cid:0) Γ (cid:15) G radΓ (cid:1) . Hence by construction of Knieper’s measure from the geodesic current µ , thedynamical system (cid:0) Γ (cid:15) G , g Γ , m Γ (cid:1) is conservative if and only if µ (cid:0) ∂ ( G \ G radΓ ) (cid:1) = 0,and it is dissipative if and only if µ ( ∂ G radΓ ) = 0.We next treat the dynamical system (cid:0) Γ (cid:15) [ G ] , g Γ , m Γ (cid:1) ; let Ω D denote its dissipativepart and Ω C its conservative part. Then again by Poincar´e’s recurrence theoremand Hopf’s divergence theorem we have m Γ (Ω D ) = m Γ (cid:0) Γ (cid:15) [ G wdivΓ ] (cid:1) and m Γ (Ω C ) = m Γ (cid:0) Γ (cid:15) [ G wrecΓ ] (cid:1) . From Lemma 6.2 we further get[
R ∩ G wdivΓ ] ⊆ [ R ∩ G \ G radΓ ] and [
R ∩ G wrecΓ ] ⊆ [ R ∩ G radΓ ] . Since [ R ] = [ R ∩ G radΓ ] ⊔ [ R ∩ G \ G radΓ ] and as the weak Ricks’ measure is supportedon Γ (cid:15) [ R ], we conclude m Γ (Ω D ) = m Γ (cid:0) Γ (cid:15) [ G \ G radΓ ] (cid:1) and m Γ (Ω C ) = m Γ (cid:0) Γ (cid:15) [ G radΓ ] (cid:1) . So by construction of the weak Ricks’ measure from the geodesic current µ defined on ∂ R , the dynamical system (cid:0) Γ (cid:15) [ G ] , g Γ , m Γ (cid:1) is conservative if and only if µ (cid:0) ∂ [ G \ G radΓ ] (cid:1) = µ (cid:0) ∂ ( G \ G radΓ ) (cid:1) = 0, and it is dissipative if and only if µ (cid:0) ∂ [ G radΓ ] (cid:1) = µ ( ∂ G radΓ ) = 0.The last statement is obvious (see the paragraph before Theorem 5.1). (cid:3) As a consequence we get the following statement which generalizes Lemma 7.5 in[35] (where the stronger assumption of a finite weak Ricks’ measure m Γ is needed): Corollary 2.
Let µ be a geodesic current defined on ∂ R . Then µ (cid:0) ∂ ( G \ G radΓ ) (cid:1) = 0 = ⇒ µ (cid:0) ∂ ( G \ G wrecΓ ) (cid:1) = 0 . Proof.
For the weak Ricks’ measure m Γ associated to the geodesic current µ theconservative part Ω C satisfies m Γ (Ω C ) = m Γ (Γ (cid:15) [ G ])according to Lemma 6.3 (b); from the proof above we further have m Γ (Ω C ) = m Γ (cid:0) Γ (cid:15) [ G wrecΓ ] (cid:1) . Hence by construction of the weak Ricks’ measure we conclude µ (cid:0) ∂ ( G \ G wrecΓ ) (cid:1) = µ (cid:0) ∂ [ G \ G wrecΓ ] (cid:1) = 0 . (cid:3) OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 17
In the sequel we will use this result to prove the necessary generalizations ofCorollary 8.3, Lemma 8.5 and Lemma 8.6 in [35], which were only proved for geo-desic currents coming from a conformal density as defined in (20), and which inducea finite
Ricks’ measure.For the remainder of this section we fix non-atomic probability measures µ − , µ + on ∂X with supp( µ ± ) = L Γ , and let µ ∼ ( µ − ⊗ µ + ) ∂ R be a quasi-product geodesic current defined on ∂ R .Notice that since the support of µ − and µ + equals L Γ , minimality of the limitset L Γ (see for example [3, Proposition 2.8]) implies that every open subset U ⊆ ∂X with U ∩ L Γ = ∅ satisfies µ ± ( U ) >
0. Hence if h ∈ Γ is a rank one element, then forthe open neighborhoods U − , U + ⊆ X of h − , h + provided by Lemma 3.1 we knowthat(11) ( µ − ⊗ µ + )( ∂ R ) ≥ µ − ( U − ) · µ + ( U + ) > µ is non-trivial. Moreover, according to the Main Theorem in [18] (see alsoProposition 6.6 (3) in [35]), the set ∂ R ∩ ( L Γ × L Γ ) is dense in L Γ × L Γ , hencesupp( µ ) = L Γ × L Γ . The first Lemma shows that in the setting of Lemma 6.3 (a) – that is when theweak Ricks’ measure associated to µ is conservative, but not necessarily finite – wehave µ ∼ µ − ⊗ µ + ; in other words we may omit the restriction to ∂ R . Lemma 6.4 (Corollary 8.3 in [35]) . If µ (cid:0) ∂ ( G \ G radΓ ) (cid:1) = 0 , then ( µ − ⊗ µ + )( ∂ R ) = ( µ − ⊗ µ + )( ∂X × ∂X ) = 1 . Proof.
From the hypothesis and Corollary 2 we get µ (cid:0) ∂ ( G \ G wrecΓ ) (cid:1) = 0 and hence(12) ( µ − ⊗ µ + ) (cid:0) ∂ ( R \ G wrecΓ ) (cid:1) = 0 . In a first step we prove that the set A := { ξ ∈ ∂X : ( ξ, η ) ∈ ∂ R for all η ∈ ∂X, η = ξ } satisfies µ − ( A ) = µ + ( A ) = 1. So let ξ ∈ ∂X be arbitrary. Our goal is to show that ξ possesses an open neighborhood U ⊆ ∂X with µ − ( U \ A ) = 0; the claim thenfollows by compactness of ∂X (and analogously for µ + instead of µ − ).Let h ∈ Γ be a rank one element. According to Lemma 3.6 (a) there exists w ∈ R with w − = ξ and w + = h + . Lemma 3.1 then provides open neighborhoods U , V ⊆ ∂X of ξ , h + such that U × V ⊆ ∂ R . From (12) we get ( µ − ⊗ µ + ) (cid:0) ( U × V ) \G wrecΓ (cid:1) = 0.For the subset W = { ζ ∈ U : ∃ u ∈ G wrecΓ such that u − = ζ, u + ∈ V } ⊆ U of U we have the inclusion ( U \ W ) × V ⊆ ( U × V ) \ G wrecΓ . Hence0 = ( µ − ⊗ µ + ) (cid:0) ( U × V ) \ G wrecΓ (cid:1) ≥ µ − ( U \ W ) · µ + ( V ) , and from µ + ( V ) > µ − ( U \ W ) = 0. As Lemma 3.11 implies W ⊆ A , weconclude µ − ( U \ A ) ≤ µ − ( U \ W ) = 0.Finally we let ξ ∈ A arbitrary. So for all η ∈ ∂X \ { ξ } we have ( ξ, η ) ∈ ∂ R .Since µ + ( { ξ } ) = 0 by non-atomicity of µ + , we have ( ξ, η ) ∈ ∂ R for µ + -almost every η ∈ ∂X . The claim then follows from µ − ( A ) = 1 and Fubini’s Theorem. (cid:3) From the previous lemma and the proof of Lemma 6.3 we immediately get
Corollary 3. µ (cid:0) ∂ ( G \ G radΓ ) (cid:1) = 0 if and only if µ − ( L radΓ ) = µ + ( L radΓ ) = 1 . For the remainder of this section we use the previous assumptions on µ − , µ + and µ ; moreover we will require that µ − ( L radΓ ) = µ + ( L radΓ ) = 1 . Lemma 6.5 (Lemma 8.5 in [35]) . Let S be any set and Ψ : ∂ R → S an arbitrarymap. If Ω ⊆ ∂ R is a set of full µ -measure in ∂ R such that for all ( ξ, η ) , ( ξ, η ′ ) , ( ξ ′ , η ′ ) ∈ Ω we have Ψ (cid:0) ( ξ, η ) (cid:1) = Ψ (cid:0) ( ξ, η ′ ) (cid:1) = Ψ (cid:0) ( ξ ′ , η ′ ) (cid:1) , then Ψ is constant µ -almost everywhere on ∂ R .Proof. From Lemma 6.4 and µ ( ∂ R \
Ω) = 0 we get( µ − ⊗ µ + )(Ω) = ( µ − ⊗ µ + )( ∂ R ) = ( µ − ⊗ µ + )( ∂X × ∂X ) . Hence for µ − -almost every ξ ∈ ∂X the set B ξ := { η ∈ ∂X : ( ξ, η ) ∈ Ω } has full µ + -measure in ∂X ; in particular, the set A := { ξ ∈ ∂X : µ + ( B ξ ) = µ + ( ∂X ) = 1 } satisfies µ − ( A ) = µ − ( ∂X ) = 1.We now fix ( ξ, η ) ∈ ( A × ∂X ) ∩ Ω. Then for any ( ξ ′ , η ′ ) ∈ ( A × B ξ ) ∩ Ω we have( ξ, η ′ ) ∈ ( A × B ξ ) ∩ Ω, hence by hypothesis on ΩΨ (cid:0) ( ξ ′ , η ′ ) (cid:1) = Ψ (cid:0) ( ξ, η ′ ) (cid:1) = Ψ (cid:0) ( ξ, η ) (cid:1) . Since the set ( A × B ξ ) ∩ Ω ⊆ ∂ R has full ( µ − ⊗ µ + )-measure in ∂X × ∂X , it alsohas full µ -measure in ∂ R . So we get Ψ (cid:0) ( ξ ′ , η ′ ) (cid:1) = Ψ (cid:0) ( ξ, η ) (cid:1) for µ -almost every( ξ ′ , η ′ ) ∈ ∂ R , and hence Ψ is constant µ -almost everywhere on ∂ R . (cid:3) The following lemma together with Lemma 3.3 is the clue to the proof of Theo-rem 6.7.
Lemma 6.6 (Lemma 8.6 in [35]) . For µ -almost every ( ξ, η ) ∈ ∂ R the isometry typeof C ( ξη ) is the same.Proof. According to Corollary 2 the set ∂ ( G wrecΓ ∩ R ) has full µ -measure in ∂ R .Moreover, if u, v ∈ G wrecΓ ∩ R satisfy u − = v − or u + = v + , then by Lemma 3.10there exist isometric embeddings between the compact metric spaces C u and C v ;hence C u and C v are isometric according to Theorem 1.6.14 in [12]. The claim nowfollows by applying Lemma 6.5 to the map which sends ( ξ, η ) ∈ ∂ ( G wrecΓ ∩ R ) to theisometry type of C ( ξη ) . (cid:3) We will now prove the appropriate generalization of Theorem 8.8 in [35], whichstates that under the additional hypothesis Z Γ = ∅ – which is satisfied in particularif X is geodesically complete – the set ∂ Z of end-point pairs of zero width geodesicshas full ( µ − ⊗ µ + )-measure in ∂X × ∂X . This will provide the key in the proof ofergodicity in Section 7. Moreover, it implies that any weak Ricks’ measure m Γ onΓ (cid:15) [ G ] associated to a quasi-product geodesic current µ ∼ ( µ − ⊗ µ + ) ∂ R is equivalentto the induced Ricks’ measure m on Γ (cid:15) G . OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 19
Theorem 6.7.
Let X be a proper Hadamard space and Γ < Is ( X ) a discrete rankone group such that Z Γ = ∅ . If µ − , µ + are non-atomic probability measures on ∂X with supp ( µ ± ) = L Γ and µ − ( L radΓ ) = µ + ( L radΓ ) = 1 , then ( µ − ⊗ µ + )( ∂ Z ) = 1 . Moreover, if µ is a quasi-product geodesic current absolutely continuous with respectto ( µ − ⊗ µ + ) ∂ R , then µ (cid:0) ∂ ( G \ Z ) (cid:1) = 0 . Proof.
By Lemma 6.6 there exists a set Ω ⊆ ∂ R of full µ -measure in ∂ R such thatthe isometry type of C ( ξη ) is the same for all ( ξ, η ) ∈ Ω. Lemma 6.4 then implies(13) ( µ − ⊗ µ + )(Ω) = 1 . Fix v ∈ Z Γ and let U − , U + ⊆ X be open neighborhoods of v − , v + accordingto Lemma 3.1. Consider decreasing sequences of open subsets ( U − n ) ⊆ U − ∩ ∂X ,( U + n ) ⊆ U + ∩ ∂X such that \ n ∈ N U − n = { v − } and \ n ∈ N U + n = { v + } . Let n ∈ N . As supp( µ ± ) = L Γ , we get ( µ − ⊗ µ + )( U − n × U + n ) = µ − ( U − n ) · µ + ( U + n ) > µ − ⊗ µ + ) (cid:0) Ω ∩ ( U − n × U + n ) (cid:1) > . So in particular there exists ( ξ n , η n ) ∈ ( U − n × U + n ) ∩ Ω.By choice of the sets U − n , U + n we get a sequence (cid:0) ( ξ n , η n ) (cid:1) ⊆ Ω ⊆ ∂ R whichconverges to ( v − , v + ) ∈ ∂ Z Γ . Now Lemma 3.4 implies that some subsequence of (cid:0) C ( ξ n η n ) (cid:1) converges, in the Hausdorff metric, to a point. As the isometry type of C ( ξη ) is the same for all ( ξ, η ) ∈ Ω, this implies that C ( ξη ) is a point for all ( ξ, η ) ∈ Ω,hence Ω ⊆ ∂ Z . We conclude( µ − ⊗ µ + )( ∂ Z ) ≥ ( µ − ⊗ µ + )(Ω) = 1 , hence µ (cid:0) ∂ ( G \ Z ) (cid:1) = 0. (cid:3) Corollary 4.
Let X be a proper Hadamard space and Γ < Is ( X ) a discrete rankone group such that Z Γ = ∅ . Let µ − , µ + be non-atomic probability measures on ∂X with supp ( µ ± ) = L Γ and µ − ( L radΓ ) = µ + ( L radΓ ) = 1 , and µ ∼ ( µ − ⊗ µ + ) ∂ R a quasi-product geodesic current defined on ∂ R . Then the weak Ricks’ measure associatedto µ is equal to the Ricks’ measure defined by (9) and also to any Knieper’s measureassociated to the quasi-product geodesic current µ (if it exists). Conservativity versus ergodicity
As before let X be a proper Hadamard space with fixed base point o ∈ X . For R > B ( R ) ⊆ G the set of all parametrized geodesic lines with origin in B o ( R ).In this section we assume that Γ < Is( X ) is a discrete rank one group with Z Γ := { v ∈ Z : v + , v − ∈ L Γ } 6 = ∅ . Notice that if X is geodesically complete, then according to Proposition 1 the lattercondition is automatically satisfied. Throughout the whole section we fix non-atomic probability measures µ − , µ + on ∂X with supp( µ ± ) = L Γ and µ − ( L radΓ ) = µ + ( L radΓ ) = 1. Let µ ∼ ( µ − ⊗ µ + ) ∂ R be a quasi-product geodesic current defined on ∂ R for which(14) ∆ := sup n ln µ (cid:0) ∂ B ( R ) (cid:1) R : R > o is finite.We next consider Ricks’ measure m associated to the geodesic current µ asdefined in (9). Since in the given setting Corollary 4 implies that Ricks’ measure isequal to weak Ricks’ measure and also to Knieper’s measure associated to the samegeodesic current µ , we will denote Ricks’ measure by m Γ instead of m . Notice thatby assumption on µ − and µ + the set G radΓ has full µ -measure; so we already knowfrom Lemma 6.3 that (Γ (cid:15) G , g Γ , m Γ ) is conservative. The goal of this section is toprove that it is also ergodic.The proof of ergodicity will make use of the famous Hopf argument (see [19], [20])as in [36] and [29], for which Theorem 6.7 is indispensable. In our more generalsetting including singular spaces we first need an analogon to Knieper’s Proposi-tion 4.1 which is valid only for manifolds. We remark that in view of Lemma 3.11our generalization of Knieper’s Proposition 4.1 is not very surprising. Lemma 7.1.
Let u ∈ Z be a Γ -recurrent rank one geodesic of zero width. Thenfor all v ∈ G with v + = u + and B v + ( v (0) , u (0)) = 0 we have lim t →∞ d ( g t v, g t u ) = 0 . Proof.
Since u is Γ-recurrent, there exist sequences ( γ n ) ⊆ Γ and ( t n ) ր ∞ suchthat γ n g t n u converges to u . Let v ∈ G be a geodesic such that v + = u + and B v + ( v (0) , u (0)) = 0. Then the function[0 , ∞ ) → [0 , ∞ ) , t d ( g t v, g t u ) = sup { e −| s | d ( v ( t + s ) , u ( t + s )) : s ∈ R } is monotone decreasing as the geodesic rays determined by u and v are asymptotic.If the function does not converge to zero as t tends to infinity, there exists a constant ǫ > d ( g t v, g t u ) ≥ ǫ for all t ≥ ǫ ≤ d ( g t n + s v, g t n + s u ) ≤ d ( v, u )for all s ≥ − t n . By Γ-invariance of d we get for all n ∈ N and for all s ≥ − t n ǫ ≤ d ( g s γ n g t n v, g s γ n g t n u ) ≤ d ( v, u ) . Passing to a subsequence if necessary we may assume that γ n g t n v converges to some v ∈ G . Hence in the limit as n → ∞ we get ǫ ≤ d ( g s v, g s u ) ≤ d ( v, u ) ≤ max { , d ( v (0) , u (0)) } for all s ∈ R . Now the first inequality shows that v = u and the second inequalitygives ( v − , v + ) = ( u − , u + ), which means that the geodesic lines v and u are parallel.Notice that in this case H o ( v ) = H o ( u ) if and only if B u − (cid:0) v (0) , u (0) (cid:1) = 0 if andonly if B u + (cid:0) v (0) , u (0) (cid:1) = 0. By choice of v we have for all n ∈ N B u + ( v ( t n ) , u ( t n )) = lim s →∞ (cid:0) d ( v ( t n ) , u ( t n + s )) − d ( u ( t n ) , u ( t n + s ) (cid:1) = lim s →∞ (cid:0) d ( γ n v ( t n ) , γ n u ( t n + s )) − s (cid:1) = lim s →∞ (cid:0) d (cid:0) ( γ n g t n v )(0) , ( γ n g t n u )( s ) (cid:1) − s (cid:1) ; OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 21 by definition of v and Γ-recurrence of u this gives0 = lim s →∞ (cid:0) d ( v (0) , u ( s )) − s (cid:1) = B u + ( v (0) , u (0)) . Hence v ∼ u which is a contradiction to v = u and u ∈ Z . (cid:3) Since we want to apply Hopf’s criterion for ergodicity Theorem 5.1 we need tofind an appropriate function ρ : Γ (cid:15) G → R in L ( m Γ ) which is strictly positive m Γ -almost everywhere. Let ∆ ≥ Lemma 7.2.
The function e ρ : G → R , u ( max { e − d ( u (0) ,γo ) : γ ∈ Γ } if u ∈ Z if u ∈ G \ Z descends to a function ρ : Γ (cid:15) G → R which is strictly positive m Γ -almost everywhereand belongs to L ( m Γ ) . Moreover, if u, v ∈ Z satisfy d (cid:0) u (0) , v (0) (cid:1) ≤ , then | e ρ ( u ) − e ρ ( v ) | ≤ e ρ ( u ) · d (cid:0) u (0) , v (0) (cid:1) . Proof.
We first notice that by definition e ρ is Γ-invariant and strictly positive on Z ,hence ρ is well-defined and strictly positive m Γ -almost everywhere (as m Γ (Γ (cid:15) Z ) = m Γ (Γ (cid:15) G ) by construction of Ricks’ measure). By definition (14) of ∆ we get m (cid:0) B ( R ) (cid:1) ≤ R · µ (cid:0) ∂ B ( R ) (cid:1) ≤ R e ∆ R . Let D Γ ⊆ G denote the Dirichlet domain for Γ with center o , that is the set of allparametrized geodesic lines with origin in { x ∈ X : d ( x, o ) ≤ d ( x, γo ) for all γ ∈ Γ } ;then for all u ∈ D Γ ∩ Z we have e ρ ( u ) = e − d ( u (0) ,o ) . Notice that if u ∈ S ( R ) := (cid:0) B ( R ) \ B ( R − (cid:1) ∩ D Γ ∩ Z , then d ( u (0) , o ) ≥ R − Z S ( R ) e ρ ( u )d m ( u ) ≤ e − R − Z B ( R ) d m ( u ) ≤ R e e − ∆ R ;this shows that ρ ∈ L ( m Γ ).We finally let u, v ∈ Z arbitrary with d (cid:0) u (0) , v (0) (cid:1) ≤
1. Let γ, γ ′ ∈ Γ such that e ρ ( u ) = e − d ( u (0) ,γo ) , e ρ ( v ) = e − d ( v (0) ,γ ′ o ) . Then e ρ ( u ) − e ρ ( v ) ≤ e − d ( u (0) ,γo ) (cid:0) − e − d ( u (0) ,v (0)) (cid:1) , e ρ ( v ) − e ρ ( u ) ≥ e − d ( u (0) ,γo ) (cid:0) e d ( u (0) ,v (0)) − (cid:1) , hence | e ρ ( u ) − e ρ ( v ) | ≤ e ρ ( u ) · max { − e − d ( u (0) ,v (0)) , e d ( u (0) ,v (0)) − }≤ e ρ ( u )2∆e d ( u (0) , v (0)) . (cid:3) For the remainder of this section we will again denote elements in the quotientΓ (cid:15) G be u, v, w and elements in G by e u, e v, e w . As we want to apply Theorem 5.1,we state the following auxiliary result. Its proof is a straightforward computationas performed in [30, page 144] using the property of e ρ stated in the last line ofLemma 7.2. Lemma 7.3.
Let f ∈ C c (Γ (cid:15) G ) be arbitrary. If u, v ∈ Γ (cid:15) Z are positively recurrentwith lifts e u , e v satisfying e u + = e v + , B e v + ( e u (0) , e v (0)) = 0 and such that f + ( u ) := lim T →∞ R T f ( g t Γ u )d t R T ρ ( g t Γ u )d t and f + ( v ) = lim T →∞ R T f ( g t Γ v )d t R T ρ ( g t Γ v )d t exist, then f + ( u ) = f + ( v ) . Proposition 2.
The dynamical system (Γ (cid:15) G , ( g t Γ ) t ∈ R , m Γ ) is ergodic.Proof. Using the last statement of Theorem 5.1 we have to show that for everyfunction f ∈ L ( m Γ ) the associated limit function f + defined by f + ( u ) := lim T →∞ R T f ( g t Γ u )d t R T ρ ( g t Γ u )d t for m Γ -almost every u ∈ Γ (cid:15) G is constant m Γ -almost everywhere; here ρ ∈ L ( m Γ ) is the function defined inLemma 7.2. As C c (Γ (cid:15) G ) is dense in L ( m Γ ) it will suffice to prove the claim for f ∈ C c (Γ (cid:15) G ).So we choose f ∈ C c (Γ (cid:15) G ) arbitrary. Since (Γ (cid:15) G , g Γ , m Γ ) is conservative, Theo-rem 5.1 states that for m Γ -almost every u ∈ Γ (cid:15) G the limits f ± ( u ) = lim T → + ∞ R T f ( g ± t Γ u )d t R T ρ ( g ± t Γ u )d t exist and are equal.As m Γ is conservative and supported on Γ (cid:15) Z , the set of recurrent elements inΓ (cid:15) Z has full measure in Γ (cid:15) G with respect to m Γ . So altogether the setΩ := { u ∈ Γ (cid:15) Z : u is positively and negatively recurrent ,f + ( u ) , f − ( u ) exist and f + ( u ) = f − ( u ) } has full measure in Γ (cid:15) G .Moreover, from the local product structure of m and Lemma 6.4 we know thatthere exists a lift e w ∈ G of some w ∈ Ω such that G e w − := { η ∈ ∂X : ∃ u ∈ Ω with a lift e u ∈ Z satisfying e u − = e w − , e u + = η } has full measure in ∂X with respect to µ + . This implies in particular that(15) m Γ (cid:0) { v ∈ Ω : ∃ lift e v ∈ Z satisfying e v + ∈ G e w − } (cid:1) = m Γ (Γ (cid:15) G ) . We will next show that f + is constant m Γ -almost everywhere on Γ (cid:15) G ; accordingto (15) above it suffices to show that for every v ∈ Ω with a lift e v ∈ Z satisfying e v + ∈ G e w − we have f + ( v ) = f + ( w ). So let v ∈ Ω be arbitrary with a lift e v ∈ Z satisfying e v + ∈ G e w − . By definition of G e w − there exists u ∈ Ω with a lift e u ∈ Z satisfying e u − = e w − and e u + = e v + ; replacing e u by g s e u for an appropriate s ∈ R ifnecessary we may further assume that B e v + ( e u (0) , e v (0)) = 0. Then the choice of e w ,the definition of Ω and Lemma 7.3 directly imply f + ( v ) = f + ( u ) = f − ( u ) . We next choose s ∈ R such that B e w − ( e w (0) , e u ( s )) = 0; from the fact that u isnegatively recurrent, e u − = e w − and Lemma 7.3 we then get f − ( w ) = f − ( g s Γ u ) . OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 23 As f ± are ( g t Γ )-invariant and w ∈ Ω, we conclude f + ( v ) = f − ( u ) = f − ( g s Γ u ) = f − ( w ) = f + ( w ) . So we have shown that m Γ -almost every v ∈ Γ (cid:15) G satisfies f + ( v ) = f + ( w ). (cid:3) We now summarize the previous results to obtain
Theorem 7.4.
Let Γ < Is ( X ) be a discrete rank one group with Z Γ = ∅ . Let µ − , µ + be non-atomic probability measures on ∂X with supp ( µ ± ) = L Γ , and µ ∼ ( µ − ⊗ µ + ) ∂ R a quasi-product geodesic current on ∂ R for which the constant ∆ ≥ defined by (14) is finite.Let m Γ be the associated Ricks’ measure on Γ (cid:15) G . Then the following statementsare equivalent: (i) µ − ( L radΓ ) = µ + ( L radΓ ) = 1 . (ii) (Γ (cid:15) G , g Γ , m Γ ) is conservative. (iii) (Γ (cid:15) G , g Γ , m Γ ) is ergodic and m Γ is not supported on a single divergent orbit.Moreover, each of the three statements implies that m Γ is equal to the weak Ricks’measure m Γ on Γ (cid:15) [ G ] and to any Knieper’s measure on Γ (cid:15) G associated to µ (if itexists). We finally mention a result concerning the dynamical systems ( ∂ G , Γ , µ ) and( ∂X × ∂X, Γ , µ − ⊗ µ + ) first introduced in Section 5. From the construction of theRicks’ measure m Γ associated to the quasi-product geodesic current µ defined on ∂ R which is absolutely continuous with respect to the product ( µ − ⊗ µ + ) ∂ R ofnon-atomic probability measures µ ± on ∂X with supp( µ ± ) = L Γ we immediatelyget Lemma 7.5. (Γ (cid:15) G , g Γ , m Γ ) is ergodic if and only if ( ∂ G , Γ , µ ) is ergodic if and onlyif ( ∂X × ∂X, Γ , µ − ⊗ µ + ) is ergodic. Geodesic currents coming from a conformal density
For the remainder of this article we will specialize to a particular kind of geodesiccurrents, namely the ones arising from a conformal density. As before X will denotea proper Hadamard space and Γ < Is( X ) a discrete rank one group. We further fixa base point o ∈ X on an invariant geodesic of a rank one element in Γ.We start with an important definition: Since Γ < Is( X ) is discrete and X isproper the orbit counting function N Γ ( R ) := { γ ∈ Γ : d ( o, γo ) ≤ R } is finite for all R >
0. The number δ Γ = lim sup R → + ∞ ln (cid:0) N Γ ( R ) (cid:1) R is called the critical exponent of Γ; it is independent of the choice of base point o ∈ X and satisfies the equality(16) δ Γ = inf { s > X γ ∈ Γ e − sd ( o,γo ) converges } . A discrete group Γ is said to be divergent if X γ ∈ Γ e − δ Γ d ( o,γo ) diverges , and convergent otherwise (that is when the infimum in (16) is attained).Given δ ≥
0, a δ -dimensional Γ-invariant conformal density is a continuous map µ of X into the cone of positive finite Borel measures on ∂X such that µ o := µ ( o ) issupported on the limit set L Γ , µ is Γ-equivariant (that is γ ∗ µ x = µ γx for all γ ∈ Γ, x ∈ X ) and(17) d µ x d µ o ( η ) = e δ B η ( o,x ) for any x ∈ X and η ∈ supp( µ o ) . The existence of a δ -dimensional Γ-invariant conformal density for δ = δ Γ goesback to S. J. Patterson ([32]) in the case of Fuchsian groups, and it turns out thathis explicit construction extends to arbitrary discrete isometry groups of Hadamardspaces with positive critical exponent (see for example [22, Lemma 2.2]). Thiscondition is satisfied for any discrete rank one group Γ < Is( X ) as it containsby definition a non-abelian free subgroup generated by two independent rank oneelements.We now fix δ > µ = ( µ x ) x ∈ X be a δ -dimensional Γ-invariant conformaldensity. By definition of a conformal density we have 0 < µ o ( ∂X ) < ∞ , and wewill assume that µ o is normalized such that µ o ( ∂X ) = 1.Before we construct a geodesic current from a conformal density we want to lista few results concerning these.We first turn our attention to the radial limit set defined by (10). Recall that for y ∈ X and r > B y ( r ) ⊆ X denotes the open ball of radius r centered at y ∈ X .If x ∈ X we define the shadow O r ( x, y ) := { η ∈ ∂X : σ x,η ( R + ) ∩ B y ( r ) = ∅} ;if ξ ∈ ∂X we set O r ( ξ, y ) := { η ∈ ∂X : ∃ v ∈ ∂ − ( ξ, η ) with v (0) ∈ B y ( r ) } . Notice that with these definitions the radial limit set can be written as L radΓ = [ c> \ R> [ γ ∈ Γ d ( o,γo ) >R O c ( o, γo );again, the definition is independent of the choice of base point o ∈ X .One corner stone result concerning δ -dimensional Γ-invariant conformal densitiesis Sullivan’s shadow lemma which gives an asymptotic estimate for the measureof the shadows O r ( o, γo ) as d ( o, γo ) tends to infinity; obviously this will lead toestimates for the measure of the radial limit set. We will need here an extension ofthe shadow lemma [26, Lemma 3.5] to the following refined versions of the shadowsabove which were first introduced by T. Roblin ([36]): For r > c > x, y ∈ X we set O + r,c ( x, y ) := { ξ ∈ ∂X : ∃ z ∈ B x ( r ) such that σ z,ξ ( R + ) ∩ B y ( c ) = ∅} , O − r,c ( x, y ) := { ξ ∈ ∂X : ∀ z ∈ B x ( r ) we have σ z,ξ ( R + ) ∩ B y ( c ) = ∅} . It is clear from the definitions that(18) O − r,c ( x, y ) = \ z ∈ B x ( r ) O c ( z, y ) ⊂ O c ( x, y ) ⊆ [ z ∈ B x ( r ) O c ( z, y ) = O + r,c ( x, y );moreover, O − r,c ( x, y ) is non-increasing in r and non-decreasing in c . We further havethe following generalization of Sullivan’s shadow lemma: Here γ ∗ µ x denotes the measure defined by γ ∗ µ x ( E ) = µ x ( γ − E ) for any Borel set E ⊆ ∂X . OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 25
Proposition 3. [29, Proposition 3 and Remark 3]
Let X be a proper Hadamardspace and Γ < Is ( X ) a discrete rank one group. Let δ > and µ a δ -dimensional Γ -invariant conformal density. Then for any r > there exists a constant c ≥ r with the following property: If c ≥ c there exists a constant D = D ( c ) > suchthat for all γ ∈ Γ with d ( o, γo ) > c we have D e − δd ( o,γo ) ≤ µ o (cid:0) O − r,c ( o, γo ) (cid:1) ≤ µ o (cid:0) O c ( o, γo ) (cid:1) ≤ µ o (cid:0) O + c,c ( o, γo ) (cid:1) ≤ D e − δd ( o,γo ) . Moreover, the upper bound holds for all γ ∈ Γ . The proof of this proposition in the special case of a Hadamard manifold X wasgiven in [29]; however the proof there does not use the fact that X is a manifold.Next we state some results from Section 3 in [26] and from Section 5 in [29]which all rely on the shadow lemma above and which remain valid in the setting ofnon-Riemannian Hadamard spaces. Lemma 8.1. [26, Proposition 3.7] If µ is a δ -dimensional Γ -invariant conformaldensity, then δ ≥ δ Γ . Lemma 8.2. [29, Lemma 5.1] If X γ ∈ Γ e − δd ( o,γo ) converges, then µ o ( L radΓ ) = 0 . In particular, if δ > δ Γ , then from (16) we immediately get µ o ( L radΓ ) = 0.Notice that the converse statement to Lemma 8.2 is much more intricate; we willhave to postpone its proof to Section 9 as we will need to work with a weak Ricks’measure on Γ (cid:15) [ G ].The following lemma states that Γ acts ergodically on the radial limit set withrespect to the measure class defined by µ : Lemma 8.3. [29, Proposition 4] If A ⊆ L radΓ is a Γ -invariant Borel subset of L radΓ ,then µ o ( A ) = 0 or µ o ( A ) = µ o ( ∂X ) = 1 . By a standard argument (see for example the proof of Theorem 4.2.1 in [30]) weget the following
Corollary 5. If µ o ( L radΓ ) > then δ = δ Γ and µ is the unique δ Γ -dimensional Γ -invariant conformal density normalized such that µ o ( ∂X ) = 1 . Finally, the following statement clarifies the possible existence of atoms:
Proposition 4. [29, Proposition 5]
A radial limit point cannot be a point mass fora δ -dimensional Γ -invariant conformal density µ . We are now going to construct a geodesic current from a δ -dimensionalΓ-invariant conformal density. Notice that according to Lemma 8.1 such a den-sity only exists if δ ≥ δ Γ .First we define for y ∈ X a map Gr y : ∂X × ∂X → R , ( ξ, η )
12 sup x ∈ X (cid:0) B ξ ( y, x ) + B η ( y, x ) (cid:1) . Obviously, the map Gr y has values in [0 , ∞ ], and comparing it to the definition byR. Ricks following [35, Lemma 5.1] we have the relation Gr y ( ξ, η ) = − β y ( ξ, η ) forall ( ξ, η ) ∈ ∂X × ∂X . Hence according to Lemma 5.2 in [35] Gr y ( ξ, η ) is finite ifand only if ( ξ, η ) ∈ ∂ G ; moreover,(19) Gr y ( ξ, η ) = 12 (cid:0) B ξ ( y, z ) + B η ( y, z ) (cid:1) if and only if z ∈ ( ξη ) lies on the image of a geodesic joining ξ and η . So the map Gr y extends the Gromov product defined in [10] via the formula (19) from ∂ G to ∂X × ∂X . By Lemma 5.3 in [35] Gr y is continuous on ∂ R and lower semicontinuouson ∂X × ∂X .We now define as in Section 7 of [35] a measure µ on ∂ G ⊆ ∂X × ∂X via(20) dµ ( ξ, η ) = e δGr o ( ξ,η ) ∂ R ( ξ, η )d µ o ( ξ )d µ o ( η ) . As ∂ G is locally compact and as µ is finite for all compact subsets of ∂ G , the measure µ is Radon; it is non-trivial by (11). Moreover, Γ-equivariance and conformality (17)of the δ -dimensional Γ-invariant conformal density µ = ( µ x ) x ∈ X occurring in theformula imply that µ is invariant by the diagonal action of Γ (and also independentof the choice of o ∈ X ).Hence as described at the end of Section 5 we can construct from the geodesiccurrent µ Knieper’s measure m Γ (provided µ is supported on ∂ Z or, more generally,if there exists a geodesic flow invariant Borel measure λ ( ξη ) on the set ( ξη ) ⊆ X for µ -almost every ( ξ, η ) ∈ ∂ G ) and both Ricks’ weak measure m Γ on Γ (cid:15) [ G ] and Ricks’measure m on Γ (cid:15) G (which will be trivial if µ ( ∂ Z ) = 0).Combining Lemma 8.2 with Lemma 6.3 (b) we get the following Proposition 5. If δ > δ Γ or if Γ is convergent, then µ ( ∂ G radΓ ) = 0 , and hencethe dynamical systems (cid:0) Γ (cid:15) G , ( g t Γ ) t ∈ R , m Γ (cid:1) with Knieper’s measure m Γ and (cid:0) Γ (cid:15) [ G ] , ( g t Γ ) t ∈ R , m Γ (cid:1) with the weak Ricks’ measure m Γ associated to µ are dissi-pative and non-ergodic unless µ is supported on a single orbit Γ · ( ξ, η ) ⊆ ∂ G . Notice that if X is a proper CAT( − < Is( X ) a non-elementarydiscrete group, then the so-called Bowen-Margulis measure (see for example [36,p.12] or [16, Section 3]) on Γ (cid:15) G – which in this case equals Γ (cid:15) Z – is preciselyKnieper’s measure m Γ or equivalently Ricks’ measure m associated to the geodesiccurrent µ .We finally mention a few further properties of the quasi-product geodesic current µ defined by (20). First, as v (0) ∈ B o ( R ) implies Gr o ( v − , v + ) ≤ R , we have µ (cid:0) ∂ { v ∈ G : v (0) ∈ B o ( R ) } (cid:1) ≤ e δR for all R >
0; hence ∆ ( ) = sup n ln µ (cid:0) ∂ B ( R ) (cid:1) R : R > o = 2 δ. Second, if µ o ( L radΓ ) = µ o ( ∂X ) = 1, then µ o is non-atomic by Proposition 4. Soaccording to Lemma 6.4 the geodesic current µ is given by(21) dµ ( ξ, η ) = e δGr o ( ξ,η ) ( ξ, η )d µ o ( ξ )d µ o ( η ) , that is the factor ∂ R in (20) can be removed. Moreover, all the equivalent state-ments of Theorem 7.4 hold.9. Conservativity in the case of divergent groups
As before, X will be a proper Hadamard space, Γ a discrete rank one group and o ∈ X a fixed base point on an invariant geodesic of a rank one element in Γ.The goal of this section is to prove the converse statement to Lemma 8.2, that is if X γ ∈ Γ e − δd ( o,γo ) diverges , then µ o ( L radΓ ) > . OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 27
However, by Lemma 8.1 a δ -dimensional Γ-invariant conformal density µ only existsif δ ≥ δ Γ ; for δ > δ Γ the Poincar´e series X γ ∈ Γ e − δd ( o,γo ) converges according to the alternative definition (16) of the critical exponent of Γ.So from here on we will assume that Γ is divergent and that µ = ( µ x ) x ∈ X is a δ Γ -dimensional Γ-invariant conformal density.In order to prove that the radial limit set of Γ has full measure with respect to µ o we follow as in [29, Section 6] Roblin’s exposition. As we want to apply thegeneralization of the second Borel-Cantelli lemma Lemma 2 in [2], we need to workwith a weak Ricks’ measure m Γ on Γ (cid:15) [ G ] and find an appropriate Borel set K ⊆ [ G ]whose projection to Γ (cid:15) [ G ] has finite m Γ -measure and which satisfies the two Renyiinequalities (27) and (28) below. Notice that in order to get a better control – anda proof even without the presence of a zero width rank one element – apart fromusing the weak Ricks’ measure we need to choose the set K more carefully than in[29, Section 6].Before we proceed we need a result concerning the following slightly refinedversion of the corridors first introduced by T. Roblin ([36]): For r > c > x, y ∈ X we set L r,c ( x, y ) := { ( ξ, η ) ∈ ∂ G : ∃ v ∈ ∂ − ( ξ, η ) ∃ t > v (0) ∈ B x ( r ) , v ( t ) ∈ B y ( c ) } . (22)Notice that in the case of a Hadamard manifold the definition is equivalent to theone given in Section 2 of [29]; however, due to the fact that the extension of ageodesic segment to a geodesic line is in general not unique in a singular Hadamardspace the definition (8) given there is not convenient here.It is clear from the definitions that L r,c ( x, y ) is non-decreasing in both r and c .Moreover, for all r ′ , c ′ > x ′ ∈ B x ( r ′ ) and y ′ ∈ B y ( c ′ ) with d ( x ′ , y ′ ) > r + r ′ + c + c ′ we have(23) L r,c ( x, y ) ⊆ L r + r ′ ,c + c ′ ( x ′ , y ′ ) , and the following result from [29] (whose proof extends to non-Riemannian Hada-mard spaces) asserts that for suitable r and c the sets L r,c ( o, γo ) are big enoughfor all but a finite number of elements in Γ. Recall that Γ < Is( X ) was assumedto be a discrete rank one group and that the base point o belongs to an invariantgeodesic of a rank one element h ∈ Γ. Proposition 6. [29, Proposition 1]
Let r > width( h ) and U − , U + ⊆ X the opendisjoint neighborhoods of h − , h + provided by Lemma 3.1 for r . Then there existsa finite set Λ ⊆ Γ such that the following holds:For any c > there exists R ≫ such that if γ ∈ Γ satisfies d ( o, γo ) > R , thenfor some β ∈ Λ we have L r,c ( o, βγo ) ∩ (cid:0) U − × U + (cid:1) ⊇ ( U − ∩ ∂X ) × O − r,c ( o, βγo ) for all r ≥ r . We fix r = r > width( h ) and open disjoint neighborhoods U − , U + ⊆ X of h − , h + provided by Lemma 3.1 for r . Let Λ ⊆ Γ be the finite subset provided byProposition 6. We then set ρ := max { d ( o, βo ) : β ∈ Λ } and – with the constant c > r from the shadow lemma Proposition 3 – fix c > c + ρ. Notice that by choice of c > r = r > width( h ) we always have c > width( h ).For this fixed constant c and with the sets U − , U + ⊆ X as above we define(24) K := { g s [ v ] : v ∈ G , v (0) ∈ B o ( c ) , ( v − , v + ) ∈ Γ( U − × U + ) , s ∈ (cid:0) − c , c (cid:1) } , which is an open subset of [ R ]. Moreover, every representative u ∈ G of [ u ] ∈ K satisfies width( u ) ≤ c : Indeed, [ u ] ∈ K implies that αu − ∈ U − and αu + ∈ U + for some α ∈ Γ; hence by Lemma 3.1 the geodesic α · u ∈ G is rank one andwidth( α · u ) ≤ c . The claim then follows from Is( X )-invariance of the widthfunction.We further remark that by construction every orbit of the geodesic flow whichenters K spends at least time c and at most time 3 c in it.In order to make the exposition of the proof of Proposition 7 below more trans-parent, we first state a few easy geometric estimates concerning intersections of theform K ∩ g − t γK and K ∩ g − t γK ∩ g − s − t ϕK in [ G ] with t, s > γ, ϕ ∈ Γ. The first one gives a relation to the sets L c,c ( o, γo )introduced in (22): Lemma 9.1. L c,c ( o, γo ) ∩ Γ( U − × U + ) ⊆ ∂ (cid:0) { K ∩ g − t γK : t > } (cid:1) ⊆ L c, c ( o, γo ) ∩ Γ( U − × U + ) Proof.
For the first inclusion we let ( ξ, η ) ∈ L c,c ( o, γo ) ∩ Γ( U − × U + ) be arbitrary.Then there exists α ∈ Γ such that ( ξ, η ) ∈ α ( U − × U + ), and by definition (22) thereexists v ∈ G with ( v − , v + ) = ( ξ, η ), d ( o, v (0)) < c and d ( γo, v ( t )) < c for some t > v ] ∈ K and, since γ − ( v − , v + ) ∈ γ − α ( U − × U + ) ⊆ Γ( U − × U + ),also γ − g t [ v ] ∈ K .For the second inclusion we let ( ξ, η ) ∈ ∂ (cid:0) { K ∩ g − t γK : t > } (cid:1) . Then ( ξ, η ) ∈ Γ( U − × U + ) and there exist v, u ∈ ∂ − ( ξ, η ), v ∼ u such that v (0) ∈ B o ( c ) and( g t u )(0) ∈ B γo ( c ) for some t >
0. Since ξ ∈ αU − and η ∈ αU + for some α ∈ Γwe know from Lemma 3.1 (since c > width( h ) and o was chosen on an invariantgeodesic of the rank one element h ) that every rank one geodesic w ∈ R joining α − ξ and α − η has width( w ) ≤ c . Now both α − v and α − u are such rank onegeodesics and therefore we get from u ∼ vd (cid:0) u ( s ) , v ( s ) (cid:1) = d (cid:0) α − u ( s ) , α − v ( s ) (cid:1) ≤ c for all s ∈ R . Choosing w ∈ G with w ∼ v such that d (cid:0) u ( s ) , w ( s ) (cid:1) = d (cid:0) w ( s ) , v ( s ) (cid:1) = 12 d (cid:0) u ( s ) , v ( s ) (cid:1) ≤ c for all s ∈ R we conclude that ( ξ, η ) = ( w − , w + ) ∈ L c, c ( o, γo ) . (cid:3) As a direct consequence we obtain that for all t, s > γ, ϕ ∈ Γ ∂ (cid:0) K ∩ g − t γK ∩ g − t − s ϕK (cid:1) ⊆ L c, c ( o, ϕo ) ∩ Γ( U − × U + ) . (25)The following geometric estimate gives a relation between the constants t, s > γ, ϕ ∈ Γ: OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 29
Lemma 9.2. K ∩ g − t γK = ∅ implies | d ( o, γo ) − t | ≤ c, and K ∩ g − t γK ∩ g − s − t ϕK = ∅ further gives ≤ d ( o, γo ) + d ( γo, ϕo ) − d ( o, ϕo ) ≤ c. Proof.
Assume that K ∩ g − t γK = ∅ . Then there exist v, u ∈ G with v ∼ u ,( v − , v + ) = ( u − , u + ) ∈ Γ( U − × U + ) and s, r ∈ ( − c/ , c/
2) such that( g s v )(0) = v ( s ) ∈ B o ( c ) and ( g r g t u )(0) = u ( r + t ) ∈ B γo ( c ) . So in particular – as in the proof of the second inclusion above – we get d (cid:0) u ( s ) , v ( s ) (cid:1) ≤ c for all s ∈ R . Hence d ( o, γo ) ≤ d ( o, v ( s )) + d ( v ( s ) , v (0)) + d ( v (0) , v ( t )) + d ( v ( t ) , u ( t ))+ d ( u ( t ) , u ( r + t )) + d ( u ( r + t ) , γo ) ≤ c + s + t + 2 c + r + c ≤ t + 5 c and similarly the reverse inequality d ( o, γo ) ≥ t − c. If K ∩ g − t γK ∩ g − s − t ϕK = ∅ , then from the first claim we get | d ( o, γo ) − t | ≤ c, | d ( o, ϕo ) − s − t | ≤ c and | d ( γo, ϕo ) − s | ≤ c. So we conclude again by the triangle inequality. (cid:3)
Finally we remark that if ( ξ, η ) ∈ L c, c ( o, ϕo ), then there exists z ∈ ( ξη ) ∩ B o (2 c )such that Gr o ( ξ, η ) = 12 (cid:0) B ξ ( o, z ) + B η ( o, z ) (cid:1) which immediately gives the estimate(26) Gr o ( ξ, η ) ≤ c. Recall that µ is a δ Γ -dimensional Γ-invariant conformal density. Let µ be thegeodesic current on ∂ G given by the formula (20) and m Γ the induced weak Ricks’measure on Γ (cid:15) [ G ] (which is supported on Γ (cid:15) [ R ]). Notice that for the projection K Γ ⊆ Γ (cid:15) [ R ] of the set K ⊆ [ R ] defined in (24) to Γ (cid:15) [ R ] we have0 < m Γ ( K Γ ) ≤ m ( K ) ≤ c · e cδ ( µ o ⊗ µ o ) (cid:0) Γ( U − × U + ) (cid:1)| {z } ≤ < ∞ . We are now going to prove the converse to Lemma 8.2 in our setting of a properHadamard space X and a discrete rank one group Γ < Is( X ). Our result heregeneralizes Proposition 1 in [29] as we neither require X to be a manifold nor Γ tocontain a strong rank one isometry or a zero width rank one isometry. Proposition 7. If P γ ∈ Γ e − δ Γ d ( o,γo ) diverges, then µ o ( L radΓ ) > .Proof. We argue by contradiction, assuming that the sum P γ ∈ Γ e − δ Γ d ( o,γo ) divergesand that µ o ( L radΓ ) = 0. We will show that for the Borel set K ⊆ [ R ] defined by(24) the following inequalities hold for T sufficiently large with universal constants C, C ′ > (27) Z T d t Z T d s X γ,ϕ ∈ Γ m ( K ∩ g − t γK ∩ g − t − s ϕK ) ≤ C (cid:18) X γ ∈ Γ d ( o,γo ) ≤ T e − δ Γ d ( o,γo ) (cid:19) (28) Z T d t X γ ∈ Γ m ( K ∩ g − t γK ) ≥ C ′ X γ ∈ Γ d ( o,γo ) ≤ T e − δ Γ d ( o,γo ) Once these inequalities are proved and under the assumption that the sum P γ ∈ Γ e − δ Γ d ( o,γo ) diverges one can apply the above mentioned generalization of thesecond Borel-Cantelli lemma, and the conclusion follows as in [36, p. 20] (applying[2, Lemma 2] to the finite measure M = m Γ restricted to K Γ ⊆ Γ (cid:15) [ R ]), namely m Γ (cid:0) { [ v ] ∈ Γ (cid:15) [ G ] : Z ∞ K Γ ∩ g − t Γ K Γ ([ v ]) = ∞} (cid:1) > . This means that the dynamical system (cid:0) Γ (cid:15) [ G ] , g Γ , m Γ (cid:1) is not dissipative. But byLemma 6.3 (b) this is a contradiction to µ o ( L radΓ ) = 0.We begin with the proof of (27): From the definition of the weak Ricks’ measureand the estimates (25) and (26) it follows that for all γ, ϕ ∈ Γ m ( K ∩ g − t γK ∩ g − t − s ϕK ) ≤ Z L c, c ( o,ϕo ) ∩ Γ( U − × U + ) d µ o ( ξ )d µ o ( η )e δ Γ Gr o ( ξ,η ) · c ≤ e cδ Γ c Z L c, c ( o,ϕo ) ∩ Γ( U − × U + ) d µ o ( ξ )d µ o ( η ) . Since obviously L c, c ( o, ϕo ) ∩ Γ( U − × U + ) ⊆ L c, c ( o, ϕo ) ⊆ ∂X × O +2 c, c ( o, ϕo ) weobtain m ( K ∩ g − t γK ∩ g − t − s ϕK ) ≤ e cδ Γ cµ o (cid:0) O +2 c, c ( o, ϕo ) (cid:1) ≤ e cδ Γ cD ( c )e − δ Γ d ( o,ϕo ) , where we used the shadow lemma Proposition 3 in the last step.Using Lemma 9.2 we finally get Z T d t Z T d s m ( K ∩ g − t γK ∩ g − t − s ϕK ) ≤ (10 c ) X γ,ϕ ∈ Γ d ( o,γo ) ≤ T +5 cd ( γo,ϕo ) ≤ T +5 c e cδ Γ cD ( c )e − δ Γ d ( o,ϕo ) ≤ c e cδ Γ D ( c ) X γ,ϕ ∈ Γ d ( o,γo ) ≤ T +5 cd ( γo,ϕo ) ≤ T +5 c e − δ Γ ( d ( o,γo )+ d ( γo,ϕo ) − c ) = 100 c e cδ Γ D ( c ) X γ,α ∈ Γ d ( o,γo ) ≤ T +5 cd ( o,αo ) ≤ T +5 c e − δ Γ ( d ( o,γo )+ d ( o,αo )) = 100 c e cδ Γ D ( c ) (cid:16) X γ ∈ Γ d ( o,γo ) ≤ T +5 c e − δ Γ d ( o,γo ) (cid:17) . Since X T
It remains to prove inequality (28). Notice first that by Lemma 3.1 every pair ofpoints ( ξ, η ) ∈ Γ( U − × U + ) can be joined by a rank one geodesic of width smallerthan or equal to twice the width of h .We recall that by construction every orbit of the geodesic flow which enters K (or one of its translates by Γ) spends at least time c in it. Using the definition of m , Lemma 9.1 and the non-negativity of the Gromov product, we first obtain for γ ∈ Γ with 5 c ≤ d ( o, γo ) ≤ T − c Z T d t m ( K ∩ g − t γK ) ≥ Z L c,c ( o,γo ) ∩ Γ( U − × U + ) d µ o ( ξ )d µ o ( η ) ≥ c z }| {(cid:18)Z c/ − c/ d s Z T d t γK (cid:0) g t ( ξ, η, s ) (cid:1)(cid:19) . Recall that r = r > width( h ) and c > c + ρ ≥ r + ρ . According to Proposition 6we know that for all γ ∈ Γ with d ( o, γo ) > R (with R > c sufficiently large) thereexists an element β in the finite set Λ ⊆ Γ with the property L r,c ( o, β − γo ) ∩ (cid:0) U − × U + (cid:1) ⊇ ( U − ∩ ∂X ) × O − r,c ( o, β − γo );using (23) and c > c + ρ ≥ r + ρ we also have the inclusion L r,c ( o, β − γo ) = β − L r,c ( βo, γo ) ⊆ β − L r + ρ,c ( o, γo ) ⊆ β − L c,c ( o, γo ) . So for all γ ∈ Γ with
R < d ( o, γo ) ≤ T − c and β = β ( γ ) ∈ Λ as above we have L c,c ( o, γo ) ∩ Γ( U − × U + ) ⊇ L c,c ( o, γo ) ∩ β ( U − × U + ) ⊇ β (cid:0) ( U − ∩ ∂X ) × O − r,c ( o, β − γo ) (cid:1) = ( βU − ∩ ∂X ) × β O − r,c ( o, β − γo )and therefore Z T d t m ( K ∩ g − t γK ) ≥ c · Z ( βU − ∩ ∂X ) × β O − r,c ( o,β − γo ) (cid:1) d µ o ( ξ )d µ o ( η )= c · µ o ( βU − ) µ o (cid:0) β O − r,c ( o, β − γo ) (cid:1) ≥ c · µ o ( βU − )e − δ Γ d ( o,β − o ) µ o ( O − r,c ( o, β − γo )) ≥ c · µ o ( βU − )e − δ Γ d ( o,β − o ) · D ( c ) e − δ Γ d ( o,β − γo ) ≥ c · min β ∈ Λ µ o ( βU − ) · e − δ Γ ρ D ( c ) e − δ Γ d ( o,γo ) = C ′′ e − δ Γ d ( o,γo ) with a constant C ′′ depending only on c and the fixed finite set Λ ⊆ Γ; in the lastthree inequalities we used the Γ-equivariance and the conformality (17) of µ , theshadow lemma Proposition 3 and the triangle inequality for the exponent. Finally, taking the sum over all elements γ ∈ Γ we get Z T X γ ∈ Γ m ( K ∩ g − t γK ) d t ≥ Z T X γ ∈ Γ R We now summarize all the previously collected results in the weakest possiblesetting: Theorem 10.1. Let X be a proper Hadamard space and Γ < Is ( X ) a discreterank one group. For δ > let µ be a δ -dimensional Γ -invariant conformal densitynormalized such that µ o ( ∂X ) = 1 , and m Γ the weak Ricks’ measure on Γ (cid:15) [ G ] as-sociated to the quasi-product geodesic current µ defined by (20). Then exactly oneof the following two complementary cases holds, and the statements (i) to (iii) areequivalent in each case:1. Case: (i) P γ ∈ Γ e − δd ( o,γo ) diverges. (ii) µ o ( L radΓ ) = 1 . (iii) (Γ (cid:15) [ G ] , g Γ , m Γ ) is conservative.2. Case: (i) P γ ∈ Γ e − δd ( o,γo ) converges. (ii) µ o ( L radΓ ) = 0 . (iii) (Γ (cid:15) [ G ] , g Γ , m Γ ) is dissipative. We remark that the first case can only happen if Γ is divergent and if δ = δ Γ . Inthis case there are several well-known additional statements: The δ Γ -dimensionalΓ-invariant conformal density µ is unique up to multiplication by a scalar. Moreoverit follows from Lemma 8.3 that µ is quasi-ergodic in the sence that every Γ-invariantBorel subset A ⊆ ∂X either has zero or full measure with respect to any measure µ x in µ . According to Proposition 4, µ is also non-atomic.Obviously, if δ > δ Γ , then we are always in the second case. Moreover, inthe second case the measure m Γ is infinite and we also have non-ergodicity of thedynamical system (Γ (cid:15) [ G ] , g Γ , m Γ ) unless the measure m Γ is supported on a singledivergent orbit { g t Γ [ v ] : t ∈ R } for some v ∈ Γ (cid:15) G ; this follows directly from theparagraph before Theorem 5.1.Since for δ > δ Γ we are always in the dissipative case we will formulate thesubsequent results only for δ = δ Γ . Under the presence of a zero width rank onegeodesic with extremities in the limit set we get the following statement whichimplies Theorem B from the introduction: OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 33 Theorem 10.2. Suppose Γ < Is ( X ) is a discrete rank one group with the extrem-ities of a zero width rank one geodesic in its limit set. Let µ be a δ Γ -dimensional Γ -invariant conformal density normalized such that µ o ( ∂X ) = 1 , and m Γ the associ-ated Ricks’ measure on Γ (cid:15) G . Then exactly one of the following two complementarycases holds, and the statements (i) to (iv) are equivalent in each case:1. Case: (i) P γ ∈ Γ e − δ Γ d ( o,γo ) diverges. (ii) µ o ( L radΓ ) = 1 . (iii) (Γ (cid:15) G , g Γ , m Γ ) is conservative. (iv) (Γ (cid:15) G , g Γ , m Γ ) is ergodic and m Γ is not supported on a single divergent orbit.2. Case: (i) P γ ∈ Γ e − δ Γ d ( o,γo ) converges. (ii) µ o ( L radΓ ) = 0 . (iii) (Γ (cid:15) G , g Γ , m Γ ) is dissipative. (iv) (Γ (cid:15) G , g Γ , m Γ ) is non-ergodic unless m Γ is supported on a single divergentorbit. Let us discuss the relation between Theorem 10.2 above and Theorem 10.1 in thecase that L Γ contains the extremities of a zero width rank one geodesic and δ = δ Γ :If Γ is divergent, then according to Theorem 7.4 the weak Ricks’ measure is equalto the Ricks’ measure. So the statements in the first case of Theorem 10.1 are onlysupplemented by the fact that the dynamical systems are ergodic.For a convergent group Γ it is well-known that there can exist many different δ Γ -dimensional Γ-invariant conformal densities. So first of all it is possible to obtainseveral distinct weak Ricks’ measures m Γ associated to different conformal densities.And even if the same δ Γ -dimensional Γ-invariant conformal density is used in theconstruction, the Ricks’ measure m Γ can be different from the weak Ricks’ measure m Γ (as it is supported on an a priori smaller set). The statements in Theorem 10.2above and Theorem 10.1 for the second case therefore apply to any (weak) Ricks’measure constructed from a δ Γ -dimensional Γ-invariant conformal density.In order to obtain Theorem C from the introduction, we have to relate our newresults to the Main Theorem in [29]. Since the measure µ on ∂ G is used in Knieper’sconstruction, Knieper’s measure coincides with Ricks’ measure on the set Γ (cid:15) Z . Asin the divergent case the support of both Knieper’s and Ricks’ measure is Γ (cid:15) Z ,the divergent case of the Main Theorem in [29] remains true under the weakerhypothesis that Γ is a discrete rank one group. By Lemma 6.3 we further getthat the equivalent conditions in the convergent case hold under the same weakercondition. So the existence of a periodic geodesic without parallel perpendicularJacobi field in Γ (cid:15) X is not a necessary hypothesis in the Main Theorem of [29] andwe immediately get Theorem C from the introduction.Finally I want to mention that for finite m Γ – the case treated in the article[35] by R. Ricks – we are always in the first case; this follows easily from the factthat finite measure spaces are conservative. Ricks further showed ([35, Theorem 4])that if X is geodesically complete, m Γ is finite and L Γ = ∂X , then (Γ (cid:15) G , g Γ , m Γ )is mixing unless X is isometric to a tree with all edge lengths in c Z for some c > [33]). We first give a criterion for the critical exponent of a divergent subgroup ofa rank one group which extends Theorem 3.2 in [33]: Proposition 8. Let X be a proper Hadamard space and Γ < Is ( X ) a discreterank one group. If H < Γ is a divergent subgroup with L H ( L Γ , then its criticalexponent satisfies δ H < δ Γ .Proof. As L H ( L Γ we may choose a point ξ ∈ L Γ \ L H . Since L H is a closed subsetof ∂X there exists an open neighborhood U ⊆ ∂X of ξ such that U ∩ L H = ∅ . As Γis a discrete rank one group, Theorem 2.8 in [3] implies the existence of a rank oneelement g ∈ Γ such that g + ∈ U . Let V − ⊆ ∂X , V + ⊆ U be small neighborhoodsof g − , g + respectively. Taking a rank one element γ ∈ Γ independent from g andmaking V − smaller if necessary we have { γ − , γ + } ∩ V − = ∅ . Using the north-southdynamics Lemma 3.6 (b) we know that for N ∈ N sufficiently large the rank oneelement e γ = g N γg − N ∈ Γhas both fixed points in V + ⊆ U . Replacing e γ by e γ M for some M ∈ N large enoughwe may further assume that e γ ( ∂X \ U ) ⊆ U and e γ − ( ∂X \ U ) ⊆ U. We now consider the free product G = H ∗ e γ < Γ; the set { h e γh e γ · · · h k − e γh k e γ : k ∈ N , h i ∈ H \ { e }} is obviously a subset of G and hence of Γ. For any s > P Γ ( s )of Γ then satisfies P Γ ( s ) = X γ ∈ Γ e − sd ( o,γo ) ≥ ∞ X k =1 X h ,...h k ∈ H \{ e } e − sd ( o,h e γh ··· e γh k e γo ) ≥ ∞ X k =1 e − skd ( o, e γo ) X h ,...h k ∈ H \{ e } e − sd ( o,h o ) e − sd ( o,h o ) · · · e − sd ( o,h k o ) = ∞ X k =1 (cid:16) e − sd ( o, e γo ) (cid:17) k · X h ∈ H \{ e } e − sd ( o,ho ) k . Since H is divergent, the sum X h ∈ H \{ e } e − sd ( o,ho ) tends to infinity as s ց δ H .Hence there exists s > δ H such thate − s d ( o, e γo ) · X h ∈ H \{ e } e − s d ( o,ho ) > s the Poincar´e series P Γ ( s ) diverges, hence δ H < s ≤ δ Γ . (cid:3) Notice that H need not be a rank one group. However, as in [33] the aboveproposition allows to produce plenty of convergent discrete rank one isometry groupsof any Hadamard space admitting a rank one isometry. The only novelty in theproof compared to the one given by M. Peign´e in [33] is the fact that the convergentsubgroup is rank one (and hence is an example for a group in which the second caseof Hopf-Tsuji-Sullivan dichotomy holds). OPF-TSUJI-SULLIVAN DICHOTOMY FOR QUOTIENTS OF HADAMARD SPACES 35 Corollary 6. Let X be a proper Hadamard space such that Is ( X ) contains twoindependent rank one elements h, g . Then there exist N, M ∈ N such that thesubgroup G of Is ( X ) generated by { g − nN h M g nN : n ∈ N } is a convergent discrete rank one group.Proof. Let U − , U + , V − , V + ⊆ X be pairwise disjoint neighborhoods of h − , h + , g − , g + .Thanks to Lemma 3.6 (b) there exist M, N ∈ N such that(29) h ± M ( V − ∪ V + ) ⊆ U ± and g ± N ( U − ∪ U + ) ⊆ V ± . This implies that G acts freely on X and hence that G is discrete; moreover, thelimit set L G of G contains the set { g − nN h − , g − nN h + : n ∈ N } , so L G is infinite. Hence according to Lemma 4.4 G is a rank one group. The limitset L H of the conjugate discrete subgroup H = g − N Gg N < Is( X ) is contained in L G and also in V − by (29). Since h + ∈ L G , h + / ∈ V − we get L H ( L G . Obviously wealso have δ H = δ G , hence the proposition above implies that H must be convergent.As conjugate groups are simultanously convergent or divergent we conclude that G is convergent. (cid:3) Notice that the isometry group of a Hadamard space X contains two indepen-dent rank one elements whenever it admits a discrete rank one subgroup. So theabove construction in particular allows to construct plenty of convergent rank onesubgroups in a given rank one discrete isometry group of X . Acknowledgements The author would like to thank Russel Ricks for answering her questions con-cerning his article [35] and for pointing out a mistake in a previous version of thisarticle. She also thanks Marc Peign´e for his comments on the preprint. 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