Tits geometry on ideal boundaries of Busemann non-positively curved space

Tits geometry on ideal b oundaries of Busemann non-p ositive ly curv e d space P .D. Andreev No v e m ber 13, 2018 1 In tro d uction In this pap er, w e cons ider Busemann non-p ositiv ely curv ed space s (shortly Busemann spaces ). The term B use m ann sp ac e was in tro duced b y B. Bow ditc h in [2], the general g eometric infor- mation on Busemann non-p ositiv ely curv ed spaces can b e found in [11]. The class of Busemann spaces con tains all C AT (0)-spaces and strongly conv ex normed spaces . When X is a complete lo cally compact C AT (0)- space , its geometry dep ends badly on the geometric b oundary at infinity ∂ ∞ X and Tits metric Td o n it. Busemann’s curv ature non-p ositivit y condition is w eak er the n Alexandrov ’s one. This leads to definite sp ecialties of the g eometry at infinit y in Bus emann spaces. Firstly , there are tw o na tural approac hes for the definition of the g eometric b oundary at infinit y . In the C AT (0)-case the t w o approac hes gives the same result, but when X is Busemann space the results can b e essen tially differen t. W e are to consider tw o differen t ideal b oundaries — horofunction (or metric) one and geo desic one. Secondly , ev en the tw o b oundaries coincide, there a r e no natural w ay to define a metric o n ∂ ∞ X with properties of Tits metric. W e pr o pose the following trick that allows using the prop erties of Tits metric without the definition of the metric itself. Note that there are t w o ke y v alues of Tits metric on ideal b oundary of C AT (0)-space. The v a lues are π and π / 2. Conditions fo r ideal p oin ts ξ , η ∈ ∂ ∞ X under whic h inequalities Td( ξ , η ) ≥ π , (1.1) Td( ξ , η ) ≤ π (1.2) and similar inequalities comparing Tits distance with π / 2 hold, can b e describ ed g eometrically without using Tits distance. In fact, mentioned inequalities can b e considered as the collection of binary relations on the b oundary ∂ ∞ X . Consequen tly , one can define analogo us collection of binary r elat io ns for Busemann space. Here w e intro duce a collection of binary relations of t yp e (1.1), (1.2) etc. W e prov e that if X is a prop er Busemann space, some prop erties of Tits metric remain true for these binary relations. W e use the nota tion of type Td( ξ , η ) ≤ π to indicate that ideal p oin ts ξ , η ∈ ∂ g X satisfy corresp onding relation in ana logy with C AT (0)-situation. One should not think that suc h notation means a comparison of metric function Td with π . The nota tion Td means only that the pair ( ξ , η ) belongs to appropriate subset of ∂ g X × ∂ g X . 1 The pap er is structured as follo ws. In Section 2, w e recall some necessary facts from Busemann non-p ositiv ely curv ed sp a ces theory . Also w e sp ecify definitions of horofunction and geo desic compactifications X h = X ∪ ∂ h X and X g = X ∪ ∂ g X . W e establish relations b etw een the compactifications. In Section 3, w e introduce t he collection of binary r elat io ns that generalize comparison the Tits distance with π on the geo desic b oundary of complete lo cally compact Busemann space. Here w e pro v e the follo wing t w o theorems generalizing kno wn prop erties of Tits distance. Theorem 3.1. L et X b e a pr op er Busemann sp ac e, and ξ , η ∈ ∂ g X ge o desic ide al p oints. If Td( ξ , η ) > π , then ther e exists a ge o desic a : R → X with ends a ( −∞ ) = η and a (+ ∞ ) = ξ . Theorem 3.2. L et X b e a pr op er Buseman n sp ac e. Given a ge o desic a : R → X with endp oints η = a ( −∞ ) and ξ = a (+ ∞ ) p assing thr ow a (0) = o , the fol lowing c onditions ar e e quivalent. 1. Td ( ξ , η ) = π ; 2. ther e exi s t hor ofunctions Φ c enter e d i n ξ and Ψ c enter e d in η , such that the interse ction of ho r ob al ls HB (Φ , o ) ∩ HB (Ψ , o ) (1.3) is unb ounde d; 3. ther e exists a norme d semiplane in X with b oundary a . Here the horoball HB (Φ , o ) = { y ∈ X | Φ( y ) ≤ Φ( o ) } is the sublev el set of the horo function Φ. In Section 4, the collection of binary relatio ns analogous to comparison o f Tits metric with v alue π / 2 is in tro duced. These relations are defined a s subsets of ∂ h X × ∂ g X . Whe n X is C AT (0)-space, the definition agrees with standard interpretation f or the inequalities of t yp e Td ≤ π / 2 etc. W e a lso pro ve t wo ve rsions of stat ement g eneralizing tr ia ngle inequalit y connecting relations Td( ξ , η ) ≤ π with relations Td([Φ] , θ ) ≤ π / 2. Am biguity of the form ulation for suc h ” triangle inequalit y” is a consequence of the relation Td([Φ] , θ ) ≤ π / 2 asymmetry . There is third p ossible w ay to formulate the ”triangle inequality ” . Coun terexample 4 .1 sho ws that in g eneral suc h third v ersion of ”triangle inequ alit y” is wrong. In Section 5, w e a pply t he relation Td([Φ] , θ ) ≤ π / 2 to study the geometry of hor o balls at infinit y . If Φ is a horofunction then corresp onding horoball at infinit y H B ∞ (Φ) can b e presen ted as the set of p oin ts ξ ∈ ∂ g X suc h that Td([Φ] , ξ ) ≤ π / 2 . W e prov e that HB ∞ (Φ) is exactly the in tersection of the b oundary ∂ g X with t he closure HB (Φ , y ) g of arbitrary horo ball H B (Φ , y ) in geo desic compactification ∂ g X . W e a lso prov e corresp onding statemen t for horospheres a t infinit y in geo desically complete prop er Busemann space X . In the case of ho r o sphe r es the inclusion HS ∞ (Φ) ⊂ HS (Φ , y ) g \ HS ( Φ , y ) can b e exact. 2 2 Busemann spaces and their ideal compactifications In this section, w e recall necessary basic facts from Busemann spaces theory and describe t wo constructions of their boundary at infinit y . W e refer the reader to [5], [6] and [11] f o r details in geometry of geo desic spaces and non-p ositiv ely curv ed spaces. Definition 2.1. Let ( X , d ) be a geo desic space. W e use notation | xy | fo r the distance d ( x, y ) b et w een p oin ts x, y ∈ X . The segmen t connecting p oints x, y ∈ X will be denoted [ xy ]. The space X is called Busemann sp ac e (Busemann non-p ositively curve d space) if for an y t wo segmen ts [ xy ] and [ x ′ y ′ ] with corresp onding affine para meterizations γ : [ a, b ] → X , γ ′ : [ a ′ , b ′ ] → X , the function D γ ,γ ′ : [ a, b ] × [ a ′ , b ′ ] → R defined b y D γ ,γ ′ ( t, t ′ ) = | γ ( t ) γ ′ ( t ′ ) | is con ve x. Equiv alently , the g eo desic space X is Busemann space if for any three p oin ts x, y , z ∈ X , the midp oin t m b et w een x and y and the midp oin t n b et w een x and z satisfy the inequalit y | mn | ≤ 1 2 | y z | . (2.1) Sev eral more statemen ts equiv alen t to the Definition 2.1 are listed in [11, Chapter 8]. It follo ws easily from the definition 2 .1 that ev ery Busemann space is contractible and ev ery t w o p oin ts x, y ∈ X are connected b y the unique segmen t [ xy ] in X . Ev ery complete geo desic c : R → X is em b edding of the real line R to X . W e call the image c ( R ) str aight line in the space X . The r ay is a geo desic c : R + → X , where R + = [0 , + ∞ ). The ra ys c, c ′ : R + → X are called c omplem ent if the map a : R → X defined as a ( t ) = c ( t ) for t ≥ 0 and a ( t ) = c ′ ( − t ) for t ≤ 0 represe nts a complete geo des ic in X . Simplest examples of Bus emann spaces are C AT (0)-spaces and normed spaces with strongly con v ex norm. Definition 2.2. The Haussdorff distance Hd( A, B ) betw een closed subsets A, B ⊂ X is Hd( A, B ) = inf { ε | A ⊂ O ε ( B ) , B ⊂ O ε ( A ) } , where O ε ( C ) = { y ∈ X | dist( y , C ) < ε } denotes the ε -neigh b ourho o d of the set C ⊂ X . Straigh t lines a, b ⊂ X are called p ar al lel if their Haussdorff distance is finite: Hd( a, b ) < + ∞ . The norme d strip in the space X is b y definition the subset in X isometric to the strip b et w een t wo straigh t lines in normed pla ne. Ev ery normed strip is b ounded by tw o pa rallel straight lines forming the b oundary of the normed strip in X . Lemma 2.1 ( W. Rino w, [12], pp. 432, 463, [2], Lemma 1.1 and remarks) . Every two p ar al lel str aight lines in Busemann sp ac e X b ound the norme d strip in X . 3 Let X b e a prop er Busemann space. Then it has tw o natural constructions of compac- tification. W e call the first compactification X g ge o desic and the second one X h hor ofunction compactification. When X is C AT (0)-space, the tw o cons t ructions giv e the same result in the follo wing sense: the iden tity map Id : X → X has con tinuation to a homeomorphism X h → X g . In general Buse mann space X the compactifications X g and X h can b e essen tially differen t. No w w e sp ecify explicit definitions. Definition 2.3. Let X b e a non-compact prop er Busem a nn space. W e sa y that geo desic rays c, d : R + → X in X are asymptotic if their Hausdorff distance is finite: Hd( c ( R + ) , d ( R + )) < + ∞ . The asymptoticit y r elatio n σ is equiv alence on the family G ( R + , X ) of geo desic ray s in X . The factorset ∂ g X = G ( R + , X ) / σ forms the set of ge o desic ide al p oints . Give n the ra y c : R + → X w e denote c (+ ∞ ) corresp onding geo desic ideal point. If x ∈ X and ξ ∈ ∂ g X , then [ xξ ] denotes the ra y from x ∈ X in the class ξ . The ray [ xξ ] alw a ys exists and is unique. W e no w define c one top olo gy on the union X g = X ∪ ∂ g X a s following. Fix a basep oin t o ∈ X . By definition, the s equence { x i } ∞ i =1 ⊂ X g con v erges to the p oint x ∈ X g in the s ense of cone top ology if lim i →∞ | ox i | = | ox | and the se quence c i : [0 , | ox i | ] → X (2.2) of naturally parameterized segmen ts (rays) [ ox i ] con v erges to the nat ur a lly parameterized seg- men t ( r ay) [ ox ]. Suc h a top ology on the set X g do es not dep end on the c hoice of the basep oin t o . The space X g with the cone top ology is compact. It is called ge o desic c omp actific ation of the space X . The set of ideal p oin ts ∂ g X forms the ge o desic ide al b oundary . The cone top ology restricted to the b oundary ∂ g X has the base of neighbourho o ds of the p oin t ξ ∈ ∂ g X consisting of op en se t s U ξ ,K = { η ∈ ∂ g X | | c ξ ( K ) c η ( K ) | < 1 } , (2.3) whic h a r e defined for all K > 0. Here c ξ , c η : R + → X are natura l parameterizations of rays c ξ = [ oξ ] and c η = [ oη ]. F rom the ot her hand, giv en arbitra r y non-compact pr o per metric space ( X , d ), its ho ro- function (metric) compactific a tion is defin ed a s following. Definition 2.4. Let C ( X , R ) b e the space o f contin uous f unction with the top olog y of uniform con v ergence on b ounded sets. Denote C ∗ ( X , R ) = C ( X , R ) / { const } a factor-space of C ( X , R ) b y the subspace of constan t s. Fix a basep oin t o ∈ X and iden tify eve ry p oin t x ∈ X with corresp onding dis tance function d x : d x ( y ) = d ( x, y ) − d ( o, x ) . The corresp ondence x → { d x + c | c = const } defines the em b edding ν : X → C ∗ ( X , R ). The space X is iden tified with its image ν ( X ) ∈ C ∗ ( X , R ). The hor ofunction c omp actific ation X h 4 is by definition the closure of the imag e ν ( X ) in C ∗ ( X , R ). The b oundary ∂ h X = X h \ X is called hor ofunction b oundary , the functions forming the b oundary ∂ h X are called hor ofunctions . Ev ery ideal p oint of the horofunction b oundary is a class of ho r ofunctions whic h differ fr o m eac h other b y constan ts. This constructiv e approac h to the horo f unction compactification of the space X is due to M. Gromo v ([7]). The em b edding ν : X → C ∗ ( X , R ) w as in tro duced also in [9][Ch. 2]. Definition 2.5. Giv en horofunction Φ o n the space X , the hor ob al l corresp onding to the p oin t y ∈ X is b y definition the sublev el set HB (Φ , y ) = { x ∈ X | Φ( x ) ≤ Φ( y ) } . (2.4) The b oundary of the horoball HB (Φ , y ) HS = ∂ HB (Φ , y ) = { x ∈ X | Φ( x ) = Φ( y ) } (2.5) is called ho r ospher e . When X is a prop er Busemann space, the compactifications in definitions 2 .3 a nd 2 .4 satisfy the inequality X g ≤ X h in the following sense. The identit y map Id X : X → X has con tinuation to con tinuous surjection π hg : X h → X g . The map π hg can b e non- injectiv e (cf. [3], [4 ]). In particular, if the p oin t ξ ∈ ∂ g X is represen ted by the ray c : R + → X suc h that c (+ ∞ ) = ξ , then its pre-imag e π − 1 hg ( ξ ) contains t he class [ β c ] ∈ ∂ h X of t he r a y Busemann function β c defined for an y y ∈ X b y the equalit y β c ( y ) = lim t →∞ ( | c ( t ) y | − t ) . Definition 2.6. The p oin t ξ ∈ ∂ g X is called r e gular , if π − 1 hg ( ξ ) is one-p oin t set π − 1 hg ( ξ ) = { [ β c ] } . It follow s easily fr o m the compactness of the space X h and Hausdorff prop ert y o f X g that the map π hg is closed: the image of any closed subset in X h is closed in X g . As a corollary , the map π hg satisfies the follo wing ”w eak op enness” prop erty: Lemma 2.2. F or an y p oint ξ ∈ ∂ g X and an y neighb ourho o d U of its pr e-image π − 1 hg ( ξ ) ⊂ ∂ h X ther e exists a neighb ourho o d V of ξ in ∂ g X such that V ⊂ π hg ( U ) . Pr o of. The image π hg ( ∂ h X \ U ) is closed in ∂ g X , hence the set V = ∂ g X \ π hg ( ∂ h X \ U ) is op en. This set is desired neigh b ourho od of ξ . R emark 2.1 . M. Rieffel in [10] in tr oduces t he no tion of metric compactification for the prop er non-compact metric space (see also [13]). This compactification is equiv alen t to t he horo function one describ ed ab o ve. The term ”horofunction compactification” w as in tr o duced b y C. W alsh in [14]. 5 3 Tits relations for the v alue π F rom no w on the spac e X will b e a non-compact prop er Busemann space. Let ra ys c, d : R + → X with common b eginning c (0) = d (0) = o represen t p oin ts ξ = c (+ ∞ ) and η = d (+ ∞ ) in the b oundary ∂ g X . It follo ws from Busemann curv ature no n- positivity and tria ngle inequalit y that the function δ o,ξ ,η ( t ) = | c ( t ) d ( t ) | 2 t is non-decreasing on R + and bounded from ab o ve by 1. Hence it has the limit δ o ( ξ , η ) = lim t → + ∞ δ o,ξ ,η ( t ) ≤ 1 . Lemma 3.1. L et r ays c, c ′ : R + → X b e asymptotic in the dir e ction ξ ∈ ∂ g X and r ays d and d ′ : R + → X b e asymptotic i n the dir e ction η ∈ ∂ g X . L et c (0) = d (0) = o a n d c ′ (0) = d ′ (0) = o ′ . Then δ o ( ξ , η ) = δ o ′ ( ξ , η ) . Pr o of. It follo ws from the me t r ic con v exit y and asymptoticit y of ra ys that | c ( t ) c ′ ( t ) | , | d ( t ) d ′ ( t ) | ≤ | oo ′ | . Hence the triangle inequality giv es | c ( t ) d ( t ) | − 2 | o o ′ | ≤ | c ′ ( t ) d ′ ( t ) | ≤ | c ( t ) d ( t ) | + 2 | oo ′ | . F or an a rbitrary ε > 0 put T = | oo ′ | ε . Then for an y t > T the inequalit y ho lds     | c ( t ) d ( t ) | 2 t − | c ′ ( t ) d ′ ( t ) | 2 t     < ε. This pro v es the claim of the lemma. R emark 3.1 . When X is C AT (0)-space, the inequ alit y δ o ( ξ , η ) < 1 (3.1) holds iff the angle b et w een ideal p oin ts ξ a nd η satisfies the inequalit y ∠ ( ξ , η ) < π . In that case we hav e the equality for Tits distance Td( ξ , η ) = ∠ ( ξ , η ) . 6 The inequalit y (3.1) has a c haracterization in terms of quasigeo desics . First recall the definition. Definition 3.1. Given n um b ers a ≥ 1 and b ≥ 0, the map f : X → Y b et w een metric spaces ( X , d X ) and ( Y , d Y ) is called ( a, b ) -quasi-isometric if fo r all x, y ∈ X the tw o-sided inequalit y holds 1 a d X ( x, y ) − b ≤ d Y ( f ( x ) , f ( y )) ≤ ad X ( x, y ) + b. (3.2) The subse t B ⊂ Y is ε -net in Y with ε > 0, if its ε -neigh b ourho o d N ε ( B ) con ta ins Y : Y ⊂ N ε ( B ). The map f : X → Y is called ( a, b ) -quasi-isometry , if it is ( a, b )-quasi-isometric and the image f ( X ) for ms ε -net in Y fo r some ε > 0. The map f is called quasi-isome tric (corre- sp ondingly quasi-isometry ) if it is ( a, b )-quasi-isometric (correspondingly ( a, b )-quasi-isometry) for some a ≥ 1 and b ≥ 0. ( a, b ) -quasige o desic in the metric space X is ( a, b )-quasi-isometric map f : I → X for some I ⊂ R . W e also call ( a, b ) -quas i g e o desic the image f ( I ) ⊂ X in this map. Let tw o ideal p oin ts ξ , η ∈ ∂ g X in the prop er Busemann space X with basep oin t o ∈ X b e giv en. Consider rays c = [ oξ ] and d = [ o, η ] with corresp onding natural parameterizations c, d : R + → X . W e define the map f : R → X b y the equalit y f ( t ) =  c ( t ) when t ≥ 0 d ( − t ) when t ≤ 0 . (3.3) Lemma 3.2. Given ide al p oints ξ , η ∈ ∂ g X , the e quality δ o ( ξ , η ) = π ho lds iff for an y ε > 0 ther e exists a numb er b ≥ 0 such that the map (3.3) is (1 + ε, b ) -quasige o desic in X . Pr o of. Ne c essity. Let δ o ( ξ , η ) = π . Since the rays c and d are geo desic, w e need only to v erify the condition (3.2) for arbitrary n umbers − s, t ∈ R , − s < 0 < t , that is for p oin ts d ( s ) and c ( t ). F rom the one hand, the triangle inequalit y g iv es | d ( s ) c ( t ) | ≤ t + s , and therefore the right inequalit y in (3.2) holds. F rom the o t her hand, the condition δ o ( ξ , η ) = 1 means that for any ε > 0 there ex ists a n um b er T > 0, suc h that | c ( t ) d ( t ) | > 2 t 1 + ε 1 for all t > T , where ε 1 satisfies to the equation 1 1 + ε = 1 − ε 1 1 + ε 1 . W e ha v e for t > T and s ≤ t | c ( t ) d ( s ) | ≥ | c ( t ) d ( t ) | − t + s > 2 t 1 + ε 1 − ( t − s )(1 + ε 1 ) 1 + ε 1 = = t + s 1 + ε 1 − ε 1 ( t − s ) 1 + ε 1 ≥ ( t + s ) 1 − ε 1 1 + ε 1 = t + s 1 + ε . 7 Similarly , | c ( t ) d ( s ) | ≥ t + s 1 + ε for s > T and t ≤ s . Since the segmen t [ − T , T ] is compact, there exists a num ber b 1 ≥ 0 suc h that, if T ≥ t ≥ s , then | c ( s ) d ( t ) | > s + t 1 + ε − b 1 . Th us, inequalit y | c ( s ) d ( t ) | > s + t 1 + ε − b 1 (3.4) holds for a ll t ≥ s . Similar arg uments sh o w, that there exists a num b er b 2 suc h that | c ( s ) d ( t ) | > s + t 1 + ε − b 2 (3.5) for all s ≥ t . T ake b = max { b 1 , b 2 } . Then we receiv e from inequalities (3.4) and ( 3.5) demanded | c ( s ) d ( t ) | > s + t 1 + ε − b. Sufficiency. Let δ o ( ξ , η ) < π and ε > 0 b e such that 1 1 + ε > δ o,ξ ,η . W e claim that there is no b ≥ 0 for whic h the map f is (1 + ε, b )-quasigeo desic. Indeed, | c ( t ) d ( t ) | ≤ 2 δ o,ξ ,η t for all t ≥ 0. Given b ≥ 0 there exists T > 0 suc h that b < 2  1 1 + ε − δ o,ξ ,η  T . Hence for a ll t ≥ T | f ( − t ) f ( t ) | ≤ 2 t 1 + ε − b, con tradicting to the left-side inequalit y in the definition of (1 + ε, b )-quasigeo desic. R emark 3.2 . The statement of the lemma do es not mean that the map (3.3) is (1 , b )-quasigeo desic for some b > 0 . The simplest countere xample can b e constructed as following. Fix a num b er α , 1 < α < 2. T ak e the Euclidean plane E 2 with co ordinates ( x, y ). The space X is constructed b y deleting from E 2 t wo con vex domains b ounded by curv es x 2 = ±| y | α and gluing tw o Eu- clidean half -planes to their places. Consider the follo wing curv e γ : R → X in X . The image γ ( R ) consists of tw o se mipara bola s x 2 = | y | α in the half-plane x > 0. The parameterization of γ is natural. Th e curv e γ satisfies the conditions of the lemma (it connects t wo ideal p oin ts 8 with angle distance π ). But the computation sho ws that there is no b > 0 suc h t ha t γ is (1 , b )-quasigeo desic. T o see this, fix b ≥ 0 and consider the f unction f ( x ) = x 2 − α ( x 2 α − 1 α − 2 b ) . It is incre a sing to + ∞ when x 2 α − 1 α > 2 b . Hence there is x 0 > 0 satisfy ing f ( x 0 ) > b 2 . Denote y 0 = x 2 /α 0 . Then x 2 0 > 2 by 0 + b 2 and q x 2 0 + y 2 0 > y 0 + b. The p oin ts A ( x 0 , − y 0 ) and B ( x 0 , y 0 ) b elong to γ . The distance b et w een them is 2 y 0 . A t the same time, the length of the path γ from A to B is greater then 2 | O A | = 2 q x 2 0 + y 2 0 > 2 y 0 + b. Hence γ is not (1 , b )-quasigeodesic. Next, we apply t he notation Td to define fiv e binary relations on ∂ g X . The notation Td( ξ , η ) denotes Tits distance b et wee n ideal p oin ts ξ , η ∈ ∂ ∞ X in C AT (0)-spaces theory . In our case the notation of t yp e Td( ξ , η ) < π is related only f or a binar y relation, not for an y metric. Definition 3.2. Let ( X , o ) b e a pointed pro p er Buse mann space. G iv en arbitrary ideal p oints ξ , η ∈ ∂ g X w e define the follo wing binary relations: • Td ( ξ , η ) < π if δ o ( ξ , η ) < π • Td ( ξ , η ) ≤ π , if for any neigh b ourho o ds U ( ξ ) and V ( η ) of this p oin ts in the sense of cone top ology on ∂ g X there exis t p oin ts ξ ′ ∈ U ( ξ ) and η ′ ∈ V ( η ) with Td( ξ ′ , η ′ ) < π ; • Td ( ξ , η ) ≥ π , if Td( ξ , η ) < π do es not hold; • Td ( ξ , η ) > π if Td( ξ , η ) ≤ π do es not hold; • Td ( ξ , η ) = π if Td( ξ , η ) ≥ π and Td( ξ , η ) ≤ π hold sim ultaneously . The definition 3 .2 is correlated with the standard definition of Tits metric when X is C AT (0). Ob viously , Td( ξ , η ) < π ⇒ Td( ξ , η ) ≤ π and Td( ξ , η ) > π ⇒ Td( ξ , η ) ≥ π . It f o llo ws directly from the definition that Td( c ( −∞ ) , c (+ ∞ ) ≥ π for an y geo desic c : R → X . Moreo v er, w e ha ve the follow ing low er semicon tinuit y prop ert y of Td-relation with resp ect to the cone top ology . If a consequence { ξ n } ⊂ ∂ g X con v erges in the sense o f cone top olog y to the p oin t ξ ∈ ∂ g X , a consequence { η n } ⊂ ∂ g X con ve rges to the p oint η ∈ ∂ g X , and if for all n ∈ N w e ha ve Td ( ξ n , η n ) ≤ π , then Td( ξ , η ) ≤ π . No w w e study some prop erties of introduced relations. 9 Lemma 3.3. Supp ose that Td( ξ , η ) < π , c = [ oξ ] and c ′ = [ oη ] . L et m t b e an arbitr ary p oint of the se gment [ c ( t ) c ′ ( t )] whe r e t > 0 . The n | om t | → ∞ wh e n t → ∞ . Pr o of. Supp ose tha t t here exists a consequence t n → ∞ for which distances | o m t n | are b ounded. Cho ose a n accum ulatio n p oin t m ∈ X for t he fa mily m t n . Suc h accum ula tion p oin t do es exist b ecause the space X is prop er. The nat ural parameterizations o f segmen ts [ m t n c ( t n )] and [ m t n c ′ ( t n )] con v erge corresp ondingly to natural par ameterizations of ray s [ mξ ] and [ mη ] uniformly on b ounded domains in R . Since eac h pair of mentioned segmen ts hav e opp osite directions in the p oint m t n , the ra ys [ mξ ] and [ mη ] are complemen t. A con tradiction. Lemma 3.4. L et c, c ′ : R → X b e r ays with c ( 0 ) = c ′ (0) = o . L et the interse ction of hor ob al ls HB ( β c , o ) ∩ HB ( β c ′ , o ) is c omp act. T h en Td( c (+ ∞ ) , c ′ (+ ∞ ) ≥ π and ther e ex i s ts a ge o desic a : R → X with a (+ ∞ ) = c (+ ∞ ) and a ( −∞ ) = c ′ (+ ∞ ) . Pr o of. Denote T = inf { t ∈ R + | HB ( β c , c ( t )) ∩ HB ( β c ′ , c ′ ( t )) 6 = ∅ } . It follo ws from the horo functions contin uit y a nd compactness of the in tersection HB ( β c , o ) ∩ H B ( β c ′ , o ) that the set M ( T ) = H B ( β c , c ( T )) ∩ HB ( β c ′ , c ′ ( t )) is non- empty . It con tains minim um p oints of the function max { β c ( x ) , β c ′ ( x ) } in the inte rsection HB ( β c , o ) ∩ HB ( β c ′ , o ). Cho ose an arbitrary p oin t m ∈ M ( T ). Then rays [ mc (+ ∞ ) ] a nd [ mc ′ (+ ∞ ) are complemen t. R emark 3.3 . The interse ction of horoballs HB ( β c , o ) ∩ HB ( β c ′ , o ) for complemen t rays c, c ′ : R + → X ma y b e compact eve n under the condition Td( c (+ ∞ ) , c ′ (+ ∞ )) = π (cf. example in [4]). If X is C AT (0), suc h situation is impossible. The t w o following theorems generalize statemen ts 1 and 3 of Prop osition 9.11 in [6]. Theorem 3.1. L et X b e a pr op er B usemann sp ac e. If Td( ξ , η ) > π , then ther e exists a ge o desic a : R → X with en d s a ( −∞ ) = ξ a nd a (+ ∞ ) = η . Pr o of. Supp ose that t he following tw o statemen ts hold sim ultaneously: A ) Td( ξ , η ) > π and B ) the straig h t line in X with endp oin ts ξ and η do es not exist. Fix a basep oin t o ∈ X , ra ys c = [ oξ ] and c ′ = [ oη ] with natural parameterizations c, c ′ : R + → X a nd arbitrar ily large n umber K > 0. The condition A ) allows to c ho ose the n um b er K suc h that | c ( K + t ) c ′ ( K ) | > K + t + 1 10 r r r r r c ( T ) c ′ ( T ) a ′ 1 b ′ 1 ξ η o Figure 1: for all t ≥ 0, and consequen tly β c ( x ) > 0 (3.6) for all x ∈ B ( c ′ ( K ) , 1). Analogously w e may assume that β c ′ ( y ) > 0 (3.7) for all y ∈ B ( c ( K ) , 1). Here β c and β c ′ are Busemann functions defined within ray s c and c ′ corresp ondingly . In suc h a choice of K w e ha ve (3.6) for a ll x ∈ B ( c ′ ( αK ) , α ) and (3.6) fo r all y ∈ B ( c ( α K ) , α ), where α ≥ 1 is arbitrary . By the curv ature non-p ositivit y , there exists a n umber T > K suc h that for a n y t ≥ T and p oin t s a, b ∈ X with | ac ( t ) | , | bc ′ ( t ) | ≤ 2, the following ho lds. If γ a : [0 , | oa | ] → X is natural parameterization of the segmen t [ x 0 a ] and γ b : [0 , | x 0 b | ] → X is natural parameterization of the segmen t [ x 0 b ], then | c ( K ) γ a ( K ) | < 1 (3.8) and | c ′ ( K ) γ b ( K ) | < 1 . (3.9) T ak e p oin t s a ′ 1 , b ′ 1 ∈ [ c ( T ) c ′ ( T )] on distances 1 from corresp onding endp oin ts c ( T ) and c ′ ( T ): | c ( T ) a ′ 1 | = | c ′ ( T ) b ′ 1 | = 1 . It ma y happ en that one o f the t w o p oints a ′ 1 or b ′ 1 b elongs to the r ay [ oξ ] or to the ray [ oη ] correspondingly . But b oth equalities a ′ 1 = c ( T − 1) and b ′ 1 = c ′ ( T − 1) 11 r r r r r r r c ( nT ) c ( T ) c ′ ( nT ) c ′ ( T ) y n,s z n,s ξ η o Figure 2: can not hold sim ultaneously , b ecause in tha t case the union of the segmen t [ a ′ 1 b ′ 1 ] with rays [ a ′ 1 ξ ] and [ b ′ 1 η ] is a straig h t line with endp oin t s ξ a nd η . If | oa ′ 1 | > | ob ′ 1 | , then a ′ 1 6 = c ( T − 1). So w e can assume (after renotatio n if necess a ry) that | o a ′ 1 | ≥ | o b ′ 1 | and a ′ 1 / ∈ [ oξ ]. D enote a 1 = a ′ 1 and b 1 the po in t of the segmen t [ b ′ 1 c ′ ( T )], on the distance τ 1 = | x 0 a 1 | from o . Next, for all natural n ≥ 2 w e define p oin ts a n and b n and num bers τ n . F or t his, consider the follo wing functions φ n , φ ′ n : [0 , | c ( nT ) c ′ ( nT ) | ] → R + . The v alues φ n ( s ) and φ ′ n ( s ) for s ∈ [0 , | c ( nT ) c ′ ( nT ) | ] are defined a s followin g. Let y n,s ∈ [ c ( nT ) c ′ ( nT )] b e the p oin t on the distance s from c ( nT ), and z n,s ∈ [ oy n,s ] b e on the distance T from o . Then φ n ( s ) and φ ′ n ( s ) are b y defi nition φ n ( s ) = | c ( T ) z n,s | , and φ ′ n ( s ) = | c ′ ( T ) z n,s | . The function φ n ( s ) is contin uous, φ n (0) = 0 and φ n ( | c ( nT ) c ′ ( nT ) | ) = | c ( T ) c ′ ( T ) | . Simi- larly , the function φ ′ n ( s ) is con tin uous, φ ′ n (0) = | c ( T ) c ′ ( T ) | and φ ′ n ( | c ( nT ) c ′ ( nT ) | ) = 0. Denote a ′ n ∈ [ c ( nT ) c ′ ( nT )] the farest p oint from c ( nT ) suc h t ha t φ n ( s ) ≤ 1 for all s ≤ | c ( nT ) a ′ | , and b ′ n the farest point from c ′ ( nT ) such that φ ′ n ( s ) ≤ 1 for all s ≥ | c ( nT ) b ′ n | . Set τ n = max {| oa ′ n | , | ob ′ n |} . If | oa ′ n | = τ n ≥ | ob ′ n | , then we redenote a n = a ′ n and denote b n the p oin t of the segmen t [ b ′ n c ′ ( nT )] with | ob n | = τ n . In that case the p oin t a n do es no t b elong to the ray [ o ξ ] and φ n ( | c ( nT ) a n | ) = 1 . (3.10) 12 F rom the other hand, if | oa ′ n | < | ob ′ n | = τ n , then a n is the po in t of the se g ment [ c ( nT ) a ′ n ] on the distance τ n from o and we redenote b n = b ′ n . In that case the p oint b n do es not b elong to the ray [ o η ] and φ ′ n ( | c ( nT ) b n | ) = 1 . (3.11) With suc h a c ho ice, the p oin ts of the segmen ts [ oa n ] and [ ob n ] on the distance T from o are separated fro m p oin ts c ( T ) and c ′ ( T ) corresp ondingly on the distance not greater then 1. Moreo v er, one of the t w o follo wing p ossibilities holds necess a rily . The consequence a n con tains an infinitely man y p oints satisfying the equalit y (3.1 0), or the consequen ce b n con tains infinitely man y p oin ts satisfying the equalit y (3 .11). F o r definitenes s, assume the first case . Notice tha t the v alues of Busemann functions β c and β c ′ are negative in p oin t s a n and b n corresp ondingly: β c ( a n ) , β c ′ ( b n ) < 0 , ho we ver the v alues β c ′ and β c are p ositiv e (cf. (3.6 ) and (3.7)): β c ′ ( a n ) , β c ( b n ) > 0 . Therefore, for eac h segmen t [ a n b n ] one can find a p oin t m n where β c ( m n ) = β ′ c ( m n ). It follo ws from the estimation | c ( nT ) c ′ ( nT ) | < 2 nT = − β c ( c ( nT )) − β c ′ ( c ′ ( nT )) that β c ( m n ) = β c ′ ( m n ) < 0 . Consequen tly , the p oin t m n b elongs to t he inters ection of t he interiors of horoba lls: m n ∈ Int( HB ( β c , o )) ∩ In t( HB ( β c ′ , o )) . (3.12) Indep ende ntly on the c hoice of the consequence m n with condition (3 .12), a ny suc h a consequenc e { m n } ∞ n =1 has an a ccu mulation p oin t m in the compact top ological space X g . There are t w o p ossibilities: 1) there exists a conseque nce { m n } ∞ n =1 satisfying the condition (3.12) with accum ulation p oin t in X : m ∈ X , or 2) eac h accum ulation point m for an y conseq uence of ty p e { m n } ∞ n =1 is infinite: m ∈ ∂ g X . In this case a ll accumulation p oints fo r all consequences { m n } ∞ n =1 where m n ∈ [ c ( nT ) c ′ ( nT )] are infinite as w ell. In part icular, all accumulation p oin ts for consequences { a n } ∞ n =1 and { b n } ∞ n =1 are infinite. The first p ossibilit y con tradicts to the condition B ). The pro of of this fact is analo g ous to that of Lemma 3.3. W e claim that the second p ossibilit y con tra dicts to the condition A ). T o prov e that, pick out a consequence n k suc h that a sub consequence o f p oin ts a n k con- v erges in the cone to p olog y to θ ∈ ∂ g X and for all k the equalit y holds φ n k ( a n k ) = 1 , 13 and sub conseq uence of p oints b n k con v erges in the cone top ology to ζ ∈ ∂ g X . By the construc- tion w e hav e θ ∈ U ξ ,K and ζ ∈ U η,K . W e will sho w that rays p = [ oθ ] and q = [ oζ ] satisfy to the condition lim t →∞ | p ( t ) q ( t ) | 2 t < 1 . F or each n ∈ N the ball B ( a n , | a n m n | ) is con tained in the horoball H B ( β c , o ), and the ball B ( b n , | b n m n | ) in the horoball HB ( β c ′ , o ). Hence | a n b n | ≤ − β c ( a n ) − β c ′ ( b n ) . Consequen tly , for eac h t ≤ τ n the follo wing holds. If y t ∈ [ oa n ], and z t ∈ [ ob n ] are p oin ts on the distance t from o , then | y t z t | ≤ − β c ( y t ) − β c ′ ( z t ) . When k → ∞ , w e ha v e in the limit | p ( t ) q ( t ) | ≤ − β c ( p ( t )) − β c ′ ( q ( t )) . W e estimate the v alue − β c ′ ( q ( t )) b y − β c ′ ( q ( t )) ≤ t. (3.13) Next, estimate the v alue − β c ( p ( t )) using the equalit y (3.10). Put z = p ( T ) = lim k →∞ z n k , | c ( n k T ) a n k | . F or each k ∈ N the p oint z n k , | c ( n k T ) a n k | b elongs to the compact in tersection o f spheres S ( o, T ) ∩ S ( c ( T ) , 1). Consequen tly z ∈ S ( o , T ) ∩ S ( c ( T ) , 1) . The function − β c when restricted to S ( o, T ) ∩ S ( c ( T ) , 1) attains its maximum in the p oin t w ∈ S ( o, T ) ∩ S ( c ( T ) , 1). The maxim um v alue is − β c ( w ) = max {− β c ( v ) | v ∈ S ( o, T ) ∩ S ( c ( T ) , 1) } < < max {− β c ( v ) | v ∈ S ( o, T ) } = T and − β c ( z ) < T . Since the function β c is con v ex, the inequalit y holds − β c ( q ( t )) < − β c ( z ) T · t (3.14) for all t > T . Com bination of t he estimations (3.13) and (3.14) giv es the ineq ua lit y | p ( t ) q ( t ) | 2 t ≤  1 + − β c ( z ) T  2 < 1 14 for all t > T . Hence Td( θ , ζ ) < π . Since the n um b er K w as chose n arbitrarily , it follo ws Td( ξ , η ) ≤ π . A con tradiction with the condition A ) . Corollary 3.1. L et X b e a pr op er B usemann sp ac e. If p oints ξ , η ∈ ∂ g X satisfy to the r elation Td( ξ , η ) > π , then ther e exist their neighb ourho o ds U ( ξ ) , U ( η ) ⊂ ∂ g X , such that any θ ∈ U ( ξ ) and ζ ∈ U ( η ) ar e c onne cte d by a str aight lin e a : R → X with end p oints a (+ ∞ ) = θ and a ( −∞ ) = ζ . Pr o of. By definition of the relat io n Td( ξ , η ) > π there exist neigh b ourho o ds U ( ξ ) , U ( η ) ⊂ ∂ g X , suc h that if θ ∈ U ( ξ ) and ζ ∈ U ( η ), then Td ( θ, ζ ) > π . Hence the claim for p oints θ and ζ follo ws from pro ven The o r em 3.1. Lemma 3.5. L et α b e strictly c onvex norme d plane and ξ , η ∈ ∂ g α b e not op p osite endp oints of any str aight line in α . T hen ther e do es not exist (1 , d ) -quasige o desic b : R → X w i th b ( t ) → ξ when t → + ∞ , b ( t ) → η when t → −∞ . Her e d ≥ 0 is arbitr ary, c onver genc e is me ant in the sense of the c one top olo gy in α g . Pr o of. Supp ose in con tra ry that the men tioned quasigeo desic do es exist. Without lost o f g en- eralit y w e ma y assume that the quasigeo desic b : R → α is con tinuous. In fact, giv en (1 , d )- quasigeo desic one can obtain con tinuous (1 , d ′ )-quasigeo desic for some d ′ ≥ d by replacing some its pieces by straigh t segmen ts with affine parameterizations. Pic k a p oin t o = b (0). It follows from the contin uit y o f the quasigeo desic b that for a ny t > 0 there exists τ ∈ R with | ob ( τ ) | = t . The definition of (1 , d )-quasigeo desic give s the inequalit y | t − τ | ≤ d. Consider p oin ts p ∈ [ oη ] and q ∈ [ oξ ] on distances | op | = | oq | = 1 from o . The distance | pq | is | pq | = 2 − ∆ < 2 (3.15) for some ∆ > 0. F rom the other hand, le t the num bers τ n < 0 a nd τ ′ n > 0 b e suc h that | ob ( τ n ) | = | ob ( τ ′ n ) | = n . Set p oin ts p n ∈ [ o b ( τ n )] and q ∈ [ oτ ′ n ] on distances | op n | = | o q n | = 1 from o . Then | p n q n | = 1 n | b ( τ n ) b ( τ ′ n ) | ≥ 1 n ( τ ′ n − τ n − d ) ≥ 2 − 3 d n . F or n > 6 d/ ∆ the inequalit y holds | p n q n | ≥ 2 − 1 2 ∆. Com bination with ( 3.15) contradicts to the con v ergence conditions b ( t ) → ξ when t → + ∞ and b ( t ) → η when t → −∞ . Definition 3.3. The norme d s e miplane in the metric sp ace X is by definition the subset in X isometric to a half-plane in the tw o-dimensional normed space. When X is Busemann space, eac h its normed semiplane is con v ex subset isometric to a half- plane in normed space with strongly con v ex norm. Give n isometry i : ¯ α → ¯ α ′ ⊂ X , where ¯ α is a ha lf-plane b ounded b y straigh t line a in the normed space V 2 , w e sa y that t he image i ( a ) b ounds the normed semiplane ¯ α ′ in X . O b viously , i ( a ) is straight line in X . 15 Let a geo des ic a bounds a normed semiplane in X . Then it follo ws from Lemma3.5 that Td( a (+ ∞ ) , a ( −∞ )) = π . The follo wing theorem giv es more complicated statemen t. Theorem 3.2. L et X b e a pr op er Buseman n sp ac e. Given a ge o desic a : R → X with endp oints ξ = a (+ ∞ ) and η = a ( −∞ ) p assing thr ow a (0) = o , the fol lowing c onditions ar e e quivalent. 1. Td ( ξ , η ) = π ; 2. ther e exi s t hor ofunctions Φ c enter e d in ξ and Ψ c enter e d in η , fo r whi c h the interse ction of ho r ob al ls HB (Φ , o ) ∩ HB (Ψ , o ) (3.16) is unb ounde d; 3. a b ounds a n orme d semiplane in X . Pr o of. 1 ⇒ 2. Supp ose tha t fo r an y tw o horofunctions Φ and Ψ cen tered in ξ and η corre- sp ondingly , the in tersection (3.16) is b ounded. The pre-images π − 1 hg ( ξ ) and π − 1 hg ( η ) are closed in ∂ h X , a nd conseq uen tly compact. W e claim that a ll in tersections (3.16) are uniformly b ounded in X : they are con tained in some ball B ( o, R ). Indeed, suppose that there exist conseque nces { [Φ n ] } ∞ n =1 ⊂ π − 1 hg ( ξ ) and { [Ψ n } ∞ n =1 ⊂ π − 1 hg ( η ) , suc h that Φ n ( o ) = Ψ n ( o ) = 0 and R n = min { R |H B (Φ n , o ) ∩ H B (Ψ n , o ) ⊂ B ( o, R ) } → ∞ . Using the compactness of t he b oundary ∂ h X w e ma y assume (passing to a sub consequences , if necess a ry), that the consequences of horof unctions Φ n and Ψ n con v erge to horofunctions Φ and Ψ corresp ondingly . They also satisfy conditions π hg ([Φ]) = ξ and π hg (Ψ) = η . Let the consequenc e { a n } ∞ n =1 ⊂ X satisfies to equality | oa n | = R n and Φ n ( a n ) , Ψ n ( a n ) ≤ 0 . Then Φ n ( x ) , Ψ n ( x ) ≤ 0 . for all x ∈ [ oa n ]. W e ma y assume that { a n } ∞ n =1 con v erges to the infinite point θ ∈ ∂ g X . In that case the natural para mete rizations of segmen ts [ oa n ] con v erge to the natural parameterization of the ra y [ oθ ]: ev ery p oin t x ∈ [ oθ ] is the limit p oin t of the consequence x n ∈ [ oa n ] with | ox n | = | ox | . Since Φ n ( x n ) , Ψ n ( x n ) ≤ 0, then for an y ε > 0 there exists a natural N , suc h tha t the inequalities Φ n ( x ) , Ψ n ( x ) ≤ ε 16 hold for all n ≥ N . W e hav e in the limit Φ( x ) , Ψ( x ) ≤ 0 . Since x is an arbitrary p oin t o f the ra y [ oθ ], w e obtain the contradiction with the b oundedness of the in tersection (3.16). Let the n um b er R > 0 b e s uch that fo r an y horofunction Φ in the class [Φ] ∈ π − 1 hg ( ξ ) and Ψ in the class [Ψ] ∈ π − 1 hg ( η ) the inclusion holds HB (Φ , o ) ∩ HB (Ψ , o ) ⊂ B ( o, R ) . F or ev ery [Φ] ∈ ∂ h X w e consider those a horofunction Φ for whic h Φ( o ) = 0. In s uc h a c hoice of horofunctions Φ and Ψ for ev ery x ∈ S ( o, 2 R ) the inequalit y holds Φ( x ) + Ψ( x ) > 0 . The contin uous function Φ + Ψ attains its maxim um in the sphere S ( o, 2 R ) and the maximal v alue is p ositiv e. Hence there are neigh b ourho o ds O Ψ (Φ) a nd O Φ (Ψ) in the space C ( X , R ) suc h that f ( x ) + g ( x ) > 0 for a n y f ∈ O Ψ (Φ) and g ∈ O Φ (Ψ) and for all x ∈ S ( o, 2 R ). Denote U Ψ (Φ) = p ( O Ψ (Φ)) ∩ ∂ h X and U Φ (Ψ) = p ( O Φ (Ψ)) ∩ ∂ h X , where p : C ( X , R ) → C ∗ ( X , R ) is a pro jection map. Fix [Φ] ∈ ∂ h X and consider a co v ering o f t he compact set π − 1 g h ( η ) ⊂ ∂ h X b y op en sets of t yp e U Φ (Ψ) for all [Ψ] ∈ π − 1 hg ( η ). Pic k a finite subcov ering: π − 1 g h ( η ) ⊂ n [ i =1 U Φ i (Ψ i ) . Denote U η (Φ) = n [ i =1 U Ψ i (Φ i ) and U Φ ( π − 1 hg ( η )) = n [ i =1 U Φ i (Ψ i ) . No w consider the co v ering of the compact set π − 1 hg ( ξ ) b y op en sets of t yp e U η (Φ) for all [Φ] ∈ ∂ h X . T aking its finite sub co v ering we get an op en set V + = N [ j =1 U η (Φ j ) ⊃ π − 1 hg ( ξ ) and op en set V − = N \ j =1 U Φ j ( η ) , with the f ollo wing prop ert y . If [Θ + ] ∈ V + and [Θ − ] ∈ V − , then for all x ∈ S ( o, 2 R ) Θ + ( x ) + Θ − ( x ) > 0 . Consequen tly , HB (Θ + , o ) ∩ HB (Θ − , o ) ⊂ B ( o, 2 R ) . (3.17) 17 F rom Lemma 2.2, there exists neighbourho o ds U + ⊂ ∂ g X of the p oin t ξ and U − ⊂ ∂ g X of the p oin t η with the fo llo wing inclusions U + ⊂ π hg ( V + ) and U − ⊂ π hg ( V − ) . Since the map π hg is con tinu o us, we ma y assume that V ± = π − 1 hg ( U ± ). The horofunction Θ + attains its minimum in the compact set (3.17) in some p oin t y 0 . The p oin t y 0 b elongs to the b oundary of the set (3.17), hence y 0 ∈ HS (Θ − , o ) and Θ − ( y 0 ) = Θ − ( o ) = max Θ − | HB (Θ + ,o ) ∩HB (Θ − ,o ) . Moreo v er, y 0 is t he minim um p oin t for the function Θ + on H S (Θ − , o ), since Θ + ( y 0 ) ≤ Θ + ( o ) Θ + ( q ) > Θ + ( o ) for a ll q ∈ HB (Θ − , o ) \ H BH (Θ + , o ). Put θ ± = π hg ([Θ ± ]) ∈ ∂ g X . Consider ra ys [ y 0 θ − ] and [ y 0 θ + ]. F or any p oin t p ∈ [ y 0 θ + ] we ha ve the inequalit y Θ − ( p ) ≤ Θ − ( y 0 ) + | p y 0 | . F rom the ot her hand, the pro jection of the p oin t p (the nearest to p p oint) to the closed con v ex set HB (Θ − , o ) is y 0 . Indeed, if z ∈ H B (Θ − , o ), then | pz | ≥ Θ + ( z ) − Θ + ( p ) ≥ Θ + ( y 0 ) − Θ + ( p ) = | py 0 | , and since any point in X ha s unique pro jection to the b ounded compact con v ex subset, hence the po in t y 0 is namely the pro jection of p . T ak e a p oin t q ∈ [ y 0 θ − ]. Denote m the in tersection p oin t of the segmen t [ pq ] with the horosphere HS (Θ − , o ). W e hav e | pq | = | pm | + | mq | ≥ | py 0 | − Θ − ( o ) = | py 0 | + | y 0 q | ≥ | pq | . No w w e conclude that m = y 0 and rays [ y 0 θ + ] and [ y 0 θ − ] complemen t eac h other up to a complete geo desic connecting θ − with θ + . This contradicts t o the relation Td( ξ , η ) ≤ π . 2 ⇒ 3. Assume tha t the inters ection (3.1 6) is not compact. Th en its closure in X g con tains infinite p oin ts: HB (Φ , o ) ∩ HB (Ψ , o ) g ∩ ∂ g X 6 = ∅ . Eac h ray [ oθ ] with θ ∈ HB (Φ , o ) ∩ HB (Ψ , o ) g ∩ ∂ g X , is con t a ined in HB (Φ , o ) ∩ HB (Ψ , o ) g ∩ ∂ g X . Moreo v er, giv en a p oint x ∈ X , the inte r section HB (Φ , x ) ∩ HB (Ψ , x ) is non-compact as w ell, and HB (Φ , x ) ∩ H B (Ψ , x ) g ∩ ∂ g X = H B (Φ , o ) ∩ H B (Ψ , o ) g ∩ ∂ g X . W e assume tha t Φ( o ) = Ψ( o ) = 0. Giv en K > 0 denote N K ( a ) the K -neigh b orho o d of the geo des ic a . The function | Φ + Ψ | is b ounded from ab o v e by 2 K on N K ( a ). W e claim that the follo wing statemen ts hold. 18 1. The restriction of Φ + Ψ to N K ( a ) attains it s minim um in some p oin t b K ; 2. fo r any minimum point b K ∈ N K ( a ) of Φ + Ψ the rays [ b K ξ ] and [ b K η ] are complemen t; 3. the function Φ + Ψ is constan t on the rays [ b K ξ ] and [ b K η ]; 4. the p oin t b K can b e c ho se n in HB (Φ , o ) ∩ HB (Ψ , o ) ∩ S ( o, K ) on the distance K from a . Pro v e t hem. Giv en arbitra ry K ′ > K , the function Φ + Ψ attains it s minimum in the compact subse t B ( o, K ′ ) ∩ N K ( a ) in some p oin t p K ′ . A t least one of the t w o rays [ p K ′ a (+ ∞ )] or [ p K ′ a ( −∞ )] in tersects the interior o f the ball B ( o , K ′ ) in the p oin t differen t from p K ′ . F or definiteness , assume that t he b eginning part o f the ray c = [ p K a (+ ∞ ) passes in the in t erio r of B ( o, K ′ ). Then Φ + Ψ is constan t in [ p K ′ a (+ ∞ ), b ecause (Φ + Ψ)( c ( t )) is con ve x, b ounded from a b ov e and non-decreasing at the initial segmen t [0 , t 0 ], whic h corresp onds to the part of the ra y in B ( o, K ′ ). Sim ila rly , Φ + Ψ is constant along the ra y [ c ( t 0 ) a ( −∞ )]. C o nse quently , p K ′ ∈ [ c ( t 0 ) a ( −∞ )], and the ray s [ p K ′ a ( −∞ )] and [ p K ′ a (+ ∞ )] are complemen t. The union a ′ p K ′ = [ p K ′ a ( −∞ )] ∪ [ p K ′ a (+ ∞ )] is the complete geo desic parallel to a . Since the sum Φ + Ψ is non-increasing on t he ray [ p K ′ θ ], we can assume that dist( a, a ′ p K ′ ) = K . The geo desic a ′ p K ′ con tains the p oin t b K ∈ a ′ p K ′ ∩ HB (Φ , o ) ∩ HB (Ψ , o ) and (Φ + Ψ)( b K ) = (Φ + Ψ)( p K ′ ) = min x ∈ B ( o,K ′ ∩ N K ( a ) (Φ + Ψ)( x )) . F or eac h K > 0 the geo desics a and a ′ K = [ b K a (+ ∞ )] ∪ [ b K a ( −∞ )] b ound the normed strip F K . It follo ws that the minim um o f the v alue Φ + Ψ in B ( o, K ′ ) ∩ N K ( a ) is the same for all K ′ > K . Hence, t he p oin t b K can b e c hosen the same for all for all K ′ > K . This pro ves all statemen ts 1 – 4 listed ab o ve. Note that the function Φ + Ψ is linear on each segmen t [ ob K ] for all K > 0. Hence, if b K , b ′ K ∈ S ( o, K ) ∩ N K ( a ) ∩ ( HB ( Φ , o ) ∩ H B (Ψ , o ) , and ν K , ν ′ K : [0 , K ] → X are the natural parameterizations of the segmen ts [ ob K ] and [ o b ′ K ] corresp ondingly , then (Φ + Ψ)( ν K ( t )) = (Φ + Ψ)( ν ′ K ( t ) for all t ∈ [0 , K ]. Let θ ∈ ∂ X b e the accum ulation p oint for the family b K when K → + ∞ . Then (Φ + Ψ)( c ( t )) = (Φ + Ψ)( ν K ( t )) for a ll K > 0 and t ∈ [0 , K ]. Here c : R + → X (correspondingly , ν K : [0 , K ] → X ) is the natural parameterization o f the ray [ o θ ] (corresp ondingly , segmen t [ ob K ]). By this reason w e ma y assume that b K = c ( K ) for all K > 0. Under suc h assumption, the normed strips F K are ordered b y inclusion: F K 1 ⊂ F K 2 , if K 1 ≤ K 2 . The union ¯ α = [ K > 0 F K 19 is the required normed semiplane. 3 ⇒ 1. By Definition 3 .2 , Td( ξ , η ) ≥ π . Supp ose that Td( ξ , η ) > π . Consider a neigh b ourho o d U + of the p oint ξ and a neighbourho o d U − of the p oint η in ∂ g X suc h that any pair of p oin ts in these neigh b ourho o ds are t he pair of endpoints for some geo desic in X . Draw ra ys [ oθ + ] and [ oθ − ] in directions of some ideal p oints θ ± ∈ U ± corresp ondingly differ from ξ and η in the normed semiplane with b oundary a . Let b : R → X b e a geo desic in X with endp oin ts θ + and θ − . The pro jection p ◦ b o f the geo desic b to the normed semiplane ¯ α b ounded b y a represen ts (1 , d )-quasigeo desic , where d = 2 max dist( b, ¯ α ). In fact, the pro jection p to the semiplane is a submetry: | p ( x ) p ( y ) | ≤ | xy | for all x, y ∈ X . Hence | ( p ◦ b )( s )( p ◦ b )( t ) | ≤ | s − t | for all s, t ∈ R . F rom the other hand, | s − t | = | b ( s ) b ( t ) | ≤ | b ( s )( p ◦ b )( s ) | + | ( p ◦ b )( s )( p ◦ b )( t ) | + | ( p ◦ b )( t ) b ( t ) | ≤ ≤ | ( p ◦ b )( s )( p ◦ b )( t ) | + 2 max dist( b, ¯ α ) , and conseque ntly | ( p ◦ b )( s )( p ◦ b )( t ) | ≥ | s − t | − d . Notice that t he map p ◦ b rep r es ents (1 , d )-quasigeo desic within the in terior metric of the semiplane ¯ α . But Lemma 3.5 states that the normed semiplane with strongly con vex norm admits no (1 , d )-quasigeodesic with endp oin ts differen t from endp oin ts of its b oundary . A con tradiction. 4 Tits relations for the v alue π / 2 Here we introduce similar collection of binary relatio ns corresp onding to the angle v alue π / 2. The follo wing lemma serv es as motiv a tion for the Definition 4.1 below. Lemma 4.1. L et X b e a pr op er C AT (0) -sp ac e, ξ , θ ∈ ∂ ∞ X . Given arbitr ary p oints x, y ∈ X c onsider r ays c = [ xξ ] and d = [ y θ ] with natur al p ar ameterizations c , d : R + → X c orr esp ond- ingly. L et β c : X → R b e a Busemann function define d by the r ay c . 1. Th e ine quality Td( ξ , η ) ≤ π / 2 holds iff the function β c ◦ d : R + → R is non-in c r e asing on R + . 2. Th e ine quality Td( ξ , η ) < π / 2 h olds iff t he function β c ◦ d : R + → R de cr e ases sublin e arly, that is if ther e ex i s t numb ers k < 0 and b ∈ R , such that for al l t ∈ R + the ine quality holds ( β c ◦ d )( t ) ≤ k t + b. Pr o of. Firstly , consider the case when x = y . In t hat case the inequality Td( ξ , η ) ≤ π / 2 is equiv alen t to the condition | c ( s ) d ( t ) | 2 ≤ s 2 + t 2 (4.1) 20 for all s, t ≥ 0. Let b e Td( ξ , η ) ≤ π / 2. Fix a num ber t ≥ 0 and start to enlarge the v alue s un b oundedly . Then the inequalit y (4.1) implies β c ( d ( t )) = lim s → + ∞ ( | c ( s ) d ( t ) | − s ) ≤ lim s → + ∞ ( √ s 2 + t 2 − s ) = 0 . Hence the function β c ◦ d is non- positive. It follow s from its con vex ity , tha t β c ◦ d is non- increasing function. Con v ersely , if the function β c ◦ d is non-increasing, then β c ( d ( t )) ≤ 0 for a ll t ≥ 0. Fix t 0 , s 0 ≥ 0. Then for an y ε > 0 there is S > s 0 suc h that for all s > S the inequalit y holds | c ( s ) d ( t ) | < s + ε. If there exists s 1 suc h that | c ( s 1 ) d ( t 0 ) | ≤ s 1 , then w e o btain fro m C AT (0)-inequalit y for the triangle △ ( xc ( s 1 ) d ( t 0 )) | c ( s 0 ) d ( t 0 ) | < s 2 0 + t 2 0 . If w e can not find suitable v alue s 1 , then for arbitr ary s 1 > S denote z the p oint of the segmen t [ c ( s 1 ) d ( t 0 ) | with | z c ( s 1 ) | = s 1 . Now it f ollo ws fr o m C AT (0)-inequality for t he triangle △ ( xc ( s 1 ) z ) and the triangle ine qualit y that | c ( s 0 ) d ( t 0 ) | < s 2 0 + t 2 0 + ε. Since the c hoice of ε > 0 for v alues s 0 , t 0 w as arbitrary , w e ha ve | c ( s 0 ) d ( t 0 ) | < s 2 0 + t 2 0 . The v alues s 0 and t 0 w as also c hosen arbitrarily , consequen tly w e hav e Td( ξ , η ) ≤ π / 2 fo r p oin ts ξ , η ∈ ∂ ∞ X . The inequalit y Td ( ξ , η ) < π / 2 is equiv alen t to the condition that there exists a n um b er λ > 0, with condition s 2 + t 2 − | c ( s ) d ( t ) | 2 2 st > λ (4.2) for all s, t > 0. The inequalit y (4.2) is equiv alen t to | c ( s ) d ( t ) | < √ s 2 + t 2 − 2 λst, and w e obtain for fix ed t > 0 that β c ( d ( t )) = lim s →∞ ( | c ( s ) d ( t ) | − s ) ≤ − λt. Finally , in the case x = y the strong inequalit y Td( ξ , η ) < π / 2 is equiv alen t to t he follo wing one | c ( s ) d ( t ) | < k t, for all s, t > 0, where the n um b er k can be tak en a s k = − λ/ 2. If the p oin ts x and y do es not coincide, consider the ra y d ′ with b eginning x asymptotic to d : d ′ = [ xθ ]. It is clear t hat the function β c ◦ d is non-increasing (corresp ondingly sublinearly decreasing) iff t he function β c ◦ d ′ is suc h. This pro ve s the lemma in the general case. 21 Lemma 4.2. L et X b e a pr op er Busemann sp ac e and Φ : X → R — hor ofunction. 1. I f for some r ay c : R + → X the function Φ ◦ c is non-incr e asing, then for any r ay c ′ : R + → X as ymp totic to c the function Φ ◦ c ′ is non -incr e a s ing as wel l. 2. I f f or s o me r ay c : R + → X the function Φ ◦ c de cr e ases subline arly, then for any r ay c ′ : R + → X as ymp totic to c the function Φ ◦ c ′ de cr e ases subline arly on R + as wel l. Pr o of. Notice that if the function f : R + → R is con ve x, it is non-increasing iff it is b ounded from ab o ve b y the v alue f (0). The con v exity of the horof unction Φ means that its restriction to an y geo desic segmen t is con v ex. If the ra y c ′ is asymptotic to c , then there are es timat io ns | c ( t ) c ′ ( t ) | ≤ | c (0 ) c ′ (0) | and Φ( c ′ ( t )) ≤ Φ( c (0)) + | c (0) c ′ (0) | . (4.3) Hence, if Φ ◦ c is non-increasing, then Φ ◦ c ′ is b ounded from ab ov e. Ob viously , the maximal v alue is (Φ ◦ c ) (0). Consequen tly , Φ ◦ c ′ is also non-increasing. Let Φ ◦ c decre a ses subline a rly: Φ( c ( t )) < k t + b for some k < 0 and b ∈ R and for a ll t ≥ 0. Then it fo llo ws from (4.3) that Φ( c ′ ( t ) < k t + b + | c (0 ) c ′ (0) | , for all t ≥ 0 and consequen tly the claim of the lemma is true. No w we are ready to define the new collection of binary relations. Eve r y one of them is formally a subset in ∂ h X × ∂ g X . W e use the notation Td again. The notation reflects the analogy with the Tits metric. Definition 4.1. Let X b e a prop er Busemann space. Given ideal p oin ts [Φ] ∈ ∂ h X a nd ξ ∈ ∂ g X we define the follow ing binary relations. 1. Td ([Φ] , ξ ) ≤ π / 2 if the horo function Φ is non- increasin g on some ra y c : R + → X with the endp oin t c (+ ∞ ) = ξ . In this case b y the Lemma 4.2 an y horofunction Φ ′ ∈ [Φ] is non-increasing on an y ra y c ′ : R + → X asymptotic to c . 2. Td ([Φ] , ξ ) < π / 2 if Φ decreases sublinearly on an y ra y c : R + → X with endp oin t c (+ ∞ ) = ξ . 3. Td ([Φ] , ξ ) > π / 2 if not Td([Φ ] , ξ ) ≤ π / 2; 4. Td ([Φ] , ξ ) ≥ π / 2 if not Td ([Φ] , ξ ) < π / 2; 5. Td ([Φ] , ξ ) = π / 2 if Td([Φ] , ξ ) ≥ π / 2 and Td([Φ] , ξ ) ≤ π / 2 sim ultaneously . 22 Geometrically the condition Td([Φ] , ξ ) > π / 2 means that rays in ξ -direction lea ve any horoball Φ ≤ const. The statemen t of the following lemma generalizes to the condition Td( ξ , η ) ≤ π / 2 in Busemann spaces the prop ert y of low-semicon tin uit y with respect to the cone top ology , known in C AT (0)-spaces case. Lemma 4.3. L et the se quenc e of hor ofunctions Ψ n c onver ges (uniformly on b ounde d subsets) to the hor ofunction Φ and se quenc e of ge o desic ide al p oints ζ n c onver ges in the c one top olo gy to the p oint ξ ∈ ∂ g X . L et Td( [Ψ n ] , ζ n ) ≤ π / 2 f o r al l natur al n . Then Td([Φ] , ξ ) ≤ π / 2 . Pr o of. Fix a basep o in t o ∈ X . Let c n , c : R + → X b e natur a l parameterizations of ray s [ oζ n ] and [ oξ ] corresp ondingly . Let n um b ers t, ε > 0 b e arbitra ry and N ∈ N b e suc h that for all n > N the following conditions hold. 1. | Φ( c ( t )) − Ψ n ( c ( t )) | < ε 2 and 2. | c ( t ) c n ( t ) | < ε 2 . Then Φ( c ( t )) < Ψ n ( c n ( t )) + ε ≤ Ψ n ( o ) + ε = Φ( o ) + ε, b ecause the horof unction Ψ n is non-increasing on the ray c n for ev ery natural n . T herefore, since ε > 0 is arbitr a ry , Φ( c ( t )) ≤ Φ( c (0)) . Since t > 0 is also arbitrary , the horofunction Φ is b ounded: Φ( c ( t )) ≤ Φ( c (0)) for all t ∈ R + . Consequen tly , Φ is non-increasing function on t he ray c and Td([Φ] , ξ ) ≤ π 2 . In cases when geo desic and metric compactifications of the space X coincide, o r when the ideal p oin t η ∈ ∂ g X is regular , w e will write Td( η , ξ ) > π / 2 a nd in the same w a y another relations from the Definition 4.1 for the class of Busemann function [ β η ] ∈ ∂ h X whic h pro jects to η . Notice that the relations are not symmetric in general: it is p ossible to b e true Td( η , ξ ) > π / 2 and Td ( ξ , η ) < π / 2 simultaneous ly . F or example, suc h pa ir s o f p oin ts can b e found in every non Euclidean normed space. Unfortunately , the author do es not kno w, whether the relation Td( ξ , η ) ≤ π / 2 implies Td( ξ , η ) < π . But there are tw o vers io ns of ”triangle inequalit y” for in tro duces relations. W e form ulate them in t w o following theorems. Theorem 4.1. L et ξ , η ∈ ∂ g X an d [Φ] ∈ ∂ h X b e id e al p oints such that Td([Φ] , ξ ) ≤ π / 2 (4.4) and Td([Φ] , η ) ≤ π / 2 . (4.5) Then Td( ξ , η ) ≤ π . 23 Pr o of. If p oin ts ξ and η are not endp oin ts for a geodesic in X , the claim of the theorem is t r ue b y Theorem 3.1. Supp ose that there exists a geo desic a : R → X with endp oin ts a ( −∞ ) = η and a (+ ∞ ) = ξ . The function Φ ◦ a is non-increasing and non-decreasing con vex function, therefore it is a constan t. W e assume Φ | a = 0. Set o = a (0). Denote θ = π hg ([Φ]) and consider a ra y c = [ oθ ] with natural parameterization c : R + → X . F or ar bit r a ry K > 0 consider the ray d ′ K = [ c ( K ) ξ ] with natural par a meteriz a tion d ′ K : R + → X . The function Φ ◦ d ′ K is non-increasing on R + b ecause of the condition (4 .4 ). But since | a ( t ) d ′ K ( t ) | ≤ | a (0) d ′ K (0) | = K fo r all t > 0, hence Φ( d ′ K ( t )) ≥ Φ( a ( t )) − K = Φ( c ( K ) ) . W e conclude that b oth the function Φ and the distance function | a ( t ) d ′ K ( t ) | are constan t on the ra y [ c ( K ) ξ ]: Φ( d ′ K ( t )) = −| a ( t ) d ′ K ( t ) | = − K . Analogously , the function Φ and the distance function | a ( − t ) d ′′ K ( t ) | are constan t on the ray d ′′ K = [ c ( K ) η ] as w ell: Φ( d ′′ K ( t )) = −| a ( − t ) d ′′ K ( t ) | = − K . W e show that the rays d ′ K and d ′′ K complemen t eac h other to the complete geo desic d K parallel to a . F or this, consider p oin ts d ′ K ( T ) and d ′′ K ( T ), where T > 0 is arbitrary and the midp oin t m of the segmen t [ d ′ K ( T ) d ′′ K ( T )]. W e ha ve the following tw o estimations for the distance | om | . At first, it follows from the metric con v exit y in the space X | om | = | a (0) m | ≤ 1 2 ( | a ( T ) d ′ K ( T ) | + | a ( − T ) d ′′ K ( − T ) | = K . F rom the other hand, since the point c ( K ) is the pro jection of o to the horoball HB (Φ , c ( K )), then | om | ≥ | oc ( K ) | = K . Consequen tly | om | = K , m = c ( K ), and therefore the union d K of ra ys d ′ K and d ′′ K is a complete geo desic. Since all its p o in ts are on the same distance K from a , w e obtain that a and d K are parallel. Denote F K the no rmed strip b et w een a and d K . It is clear that F K 1 ⊂ F K 2 when K 1 < K 2 . Therefore, the unio n ¯ α = [ K > 0 F K is a normed semiplane in X with bo undar y a . The relation Td( ξ , η ) = π follo ws no w f r om the Theorem 3.2. Theorem 4.2. L et the ide al p oints [Φ] , [Ψ] ∈ ∂ h X an d the ide al p oint ζ ∈ ∂ g X b e such that Td([Φ] , ζ ) ≤ π / 2 (4.6) and Td([Ψ] , ζ ) ≤ π / 2 . (4.7) L et ξ = π hg ([Φ]) and η = π hg ([Ψ]) . Then Td( ξ , η ) ≤ π . 24 Pr o of. Supp ose that endp oin ts o f some geo desic a : R → X are a (+ ∞ ) = ξ and a ( −∞ ) = η . Denote a (0) = o and b = [ oζ ]. By the condition, restrictions of functions Φ and Ψ to b are non- increasing. Hence the interse ction of horo ba lls HB (Φ , o ) ∩ HB (Ψ , o ) is non- compact. Applying Theorem 3.2, w e get that a b ounds a normed semiplane in X and Td( ξ , η ) = π . If the geo desic a do es not exist, the inequalit y Td( ξ , η ) ≤ π is the corollary of the Theorem 3.1. A priori o ne can formulate another v ersion f or the triangle inequalit y: Td( ξ , ζ ) ≤ π / 2 , Td( ζ , η ) ≤ π / 2 ⇒ Td( ξ , η ) ≤ π ? The fo llo wing coun terexample sho ws that the third v ersion of the tria ngle inequalit y is not correct. Coun terexample 4.1. Consider three items α 1 , α 2 and α 3 of normed semiplane with coo rdi- nates ( x 1 , x 2 ), x 2 ≥ 0 and norm k ( x 1 , x 2 ) k = 4 p x 4 1 + x 4 2 Glue them to the metric space with interior metric in the follow ing wa y: the p ositiv e b oundary ra y x 1 of the semiplane α 1 is glued to the negativ e b oundary ray − x 1 of the semiplane α 2 , the p ositiv e b oundary ray x 1 of the semiplane α 2 is glued to the negative b oundary ra y − x 1 of the semiplane α 3 and finally , the p ositiv e b oundary ray x 1 of the semiplane α 3 is glued to the negative b oundary ray − x 1 of the semiplane α 1 . The resulting space X is Busemann space. Its geo desic and horofunction compactifications coincide: the surjection π hg : X h → X g is a homeomorphism. There are ideal p oin t s ξ , η , ζ ∈ ∂ g X , suc h that Td([ β ξ ] , ζ ) < π / 2 , Td([ β ζ ] , η ) < π / 2, but Td( ξ , η ) > π . F or example, suc h p oin ts ar e infinite p oin ts of the fo llo wing ra ys: the p oin t ξ on the ray directed by the ve ctor (cos 5 π 6 , sin 5 π 6 ) in the semiplane α 3 , the p oin t ζ on the ray with directing v ector (cos π 3 , sin π 3 ) in the semiplane α 1 and the p oin t η o n the ray with directing v ector (cos( 5 π 6 + ε ) , sin( 5 π 6 + ε )), where 0 < ε < arctan  3 3 4  − π 3 in the se miplane α 1 . 5 Horoballs at infinit y Definition 5.1. Let Φ : X → R b e a horofunction that generates an ideal p oin t [Φ] ∈ ∂ h X . The hor ob al l at infinity with c enter [Φ] is b y definition a set HB ∞ (Φ) = n ξ ∈ ∂ g X | T d ([Φ] , ξ ) ≤ π 2 o . Corresp ondingly , the se t hb ∞ (Φ) = n ξ ∈ ∂ g X | Td([Φ] , ξ ) < π 2 o is called op en hor ob al l at infinity , and t he set HS ∞ (Φ) = n ξ ∈ ∂ g X | Td([Φ] , ξ ) = π 2 o is hor ospher e at infinity with cen ter [Φ]. 25 R emark 5.1 . In general the set hb ∞ (Φ) is not the interior for H B ∞ (Φ) in the sens e of the cone top ology and HS ∞ (Φ) is not it s b oundary . These statemen ts are false ev en with resp ect to Tits metric when X is C AT (0)- space . The reason of suc h effect is: the closed ball in Tits metric can be a comp onen t of linear connec t io n. In this case it will b e open set. F or example, consider the space X obta ined b y g luing by the bo undary ℓ of the Euclidean semiplane ¯ α and Lobac hevskii semiplane ¯ β . F or infinite p oint ξ ∈ ∂ ∞ X o n the ray p erp endicular to ℓ in the Euclidean semiplane, the horoball at infinity H B ∞ ( β ξ ) coincides with the b oundary ∂ ∞ ( ¯ α ), a nd the horosphere a t infinity HS ∞ ( β ξ ) with ∂ ∞ ( ℓ ). The horo ball HB ∞ ( β ξ ) = ∂ ∞ ( ¯ α ) is op en set in Tits metric. No w we giv e another description for horoballs and horospheres at infinit y in the prop er Busemann space X . Ev ery horoball HB (Φ , y ) is a sublev el set (2.4) in X , and the horosphere HS (Φ , y ) is corresp onding lev el set (2.5). When X is contained in X g with cone t o polo g y , all subsets in X hav e their closures in X g . Lemma 5.1. 1. L et X b e a pr op er Busemann sp ac e. Then HB (Φ , y ) g = HB (Φ , y ) ∪ HB ∞ (Φ) (5.1) 2. L et X b e a ge o desic al ly c omplete pr op er Busemann sp ac e. T h en HS ∞ (Φ) ⊂ HS (Φ , y ) g \ H S (Φ , y ) (5.2) Pr o of. 1. F rom the definition of horoball a t infinit y , the inclusion ξ ∈ HB ∞ ([Φ]) holds iff the horo function Φ is non-increasing at the ra y [ y ξ ]. Equiv alen tly , [ y ξ ] ⊂ H B (Φ , y ). Consequen tly the equalit y (5.1). 2. It follows from the definition of the horosphere at infinity that the inclusion ξ ∈ HS ∞ ([Φ]) holds iff t he horofunction Φ is non-increasing and do es not decreases sublinearly on the ra y [ y ξ ]. In that case the ray [ y ξ ] b elongs to the horoball HB (Φ , y ) but may pass strongly in the in terior of the hor o ball HB (Φ , y ) and b e not a subset of H S (Φ , y ). W e sho w that in an y case the p oin t ξ b elongs to the closure of the hor o sphe r e HS (Φ , y ) in ∂ g X as we ll. Let c : R + → X b e the para meteriz a tion of the ra y [ y ξ ]. Let x ∈ X b e a p oin t, suc h that Φ( x ) > Φ( y ). F or an y t > 0 there exists a p oin t z t in the horosphere HS (Φ , x ) with | z t c ( t ) | = Φ( z t ) − Φ( c ( t )) = Φ( x ) − Φ( c ( t )) . This p oint z t b elongs to the in tersection of the ho rosphere HS (Φ , x ) with arbitra r y ra y from c ( t ) in the direction o pposite to the ray [ c ( t ) π hg ([Φ])]. The v alue | y z t | ≥ t − | z t c ( t ) | is non decreasing when t → ∞ , but the difference Φ( x ) − Φ( c ( t )) increases sublinearly: for an y k > 0 and b ∈ R there exis ts T = T ( k , b ) > 0 suc h tha t Φ( x ) − Φ( c ( t )) < k t + b for all t > T . Therefore lim t →∞ Φ( x ) − Φ( c ( t )) t = 0 . 26 An y ray with start segmen t [ y z t ] pa sses out of the horoball HB (Φ , y ) and its endp oin t lays out of HB ∞ ([Φ]). A t the same time, for an y cone neighbourho o d U of the p oin t ξ there exists sufficien tly large T > 0 , suc h that endp oin ts of rays with b eginning part [ y z t ] b elong to U for all t > T . This means that U has non empty in tersection with the complemen t to the closure of the horoball H B (Φ , y ) in ∂ g X . This pro v es the inclusion ξ ∈ HS (Φ , y ) g . R emark 5.2 . The in v erse inclusion to (5.2) can b e fa lse. The simples t example is Lobac hevskii space, where all ho rosphere s at infinity a re empt y but ev ery horosphere of the s pa ce has an a c- cum ulation p oin t at infinit y — its cente r. The geo desic completeness condition is essen tial here: it is easy to construct the situation when a ll leve l sets for a horofunction in non geo desically complete space are b ounded, but the horo sphe re at infinit y is not empt y . F or example, consider the horofunction Φ( x, y ) = y on the par t y ≥ p | x | o f the Euclidean plane with co ordinates ( x, y ) R emark 5.3 . It fo llo ws f r om the Lemma 5.1 that all horoballs as sublev el sets for the horofunc- tion Φ hav e the same boundary at infinit y ∂ ∞ ( HB (Φ , x ))) = HB (Φ , x ) g \ H S (Φ , x ) . Consequen tly , the horo ball a t infinit y HB ∞ ([Φ]) can b e defined by the equalit y (5.1). In another w ords, the horoball at infinity HB ∞ (Φ) is the inv erse limit for the system of closures HB (Φ , t ) g with inclusions HB (Φ , t 1 ) g ⊂ HB (Φ , t 2 ) g when t 1 ≤ t 2 under t → −∞ . The statemen t of the follow ing theorem is another fo r mulation of the Lemma 4.3 in terms of horoballs at infinity . Theorem 5.1. L et the c onse quenc e of hor ofunctions { Φ n } ∞ n =1 c onver ges in c omp act-op en top ol- o gy to the h o r o f unc tion Φ and the p oint ξ ∈ ∂ g X is the limit of the c onse quenc e { ξ n } ∞ n =1 ⊂ ∂ g X in the sense of the c one top olo gy in ∂ g X , wher e ξ n ∈ HB ∞ (Φ n ) . (5.3) Then ξ ∈ HB ∞ (Φ) . R emark 5.4 . The claim of the The o rem 5.1 is equiv alen t the inclusion lim n →∞ HB ∞ (Φ n ) ⊂ HB ∞  lim n →∞ Φ n  , (5.4) under the condition that horofunctions Φ n con v erge to the horofunction Φ. Here the limit lim n →∞ HB ∞ (Φ n ) is the union of accum ulation p oin ts for all differen t sequences { ξ n } ∞ n =1 , where ξ n ∈ HB ∞ (Φ n ), con ve rging in the sense of the cone top ology on ∂ g X . The example describ ed in the Remark 5.2 sho ws that the inclusion (5.4) can b e strict. Ac knowledgem ent. The a uthor is ve ry grateful to Sergey Vladimiro vic h Buy alo for his atten tion to the first v ersion of the pap er and the n umber of imp ortan t remarks and corrections. 27 References [1] H. Busemann, Sp ac es with nonp ositive curvatur e , Acta Mathematica, 80 (1948), 259–310. [2] B. H. Bo wditch , Minkowskian subsp ac es of non-p ositively curve d metric sp a c es , Bull. L o n- don Math. Soc., 27 (1995), pp. 575–584. [3] P .D . Andreev, Ge ometry of ide al b oundaries of ge o desic sp ac es with nonp ositive curvatur e in B usemann sense , Sib. Adv. Math., 18 , 2 (2008). [4] P .D . Andreev, I d e al closur es of Busemann sp ac e and s i n gular Minkow ski sp ac e , http://arxi v.org/abs/math/0405121 [5] D . Burago, Y u. Burago, S. Iv ano v, A c ourse in metric ge ometry. , G raduate Studies in Mathematics. 33., 2001. [6] M. Bridson, A. Ha efliger, Metric sp ac es of non-p ositive curvatur e , Comprehe nsive Studies in Mathematics, v ol. 319, Springer-V erlag, Berlin, 1999. [7] M. Gro mo v, Hyp erb olic manifolds, gr oups and actions , in Riemann surfac es and r elate d top- ics: Pr o c e e dings of 1978 Stony Br o ok c onfer enc e , 1 8 3–213, Princ eton Univ. Press, Prince- ton, N.J., 1981. [8] Ph. Hotc hkiss, The b oundary of Busem ann sp ac e , Pro c. Amer. Math. So c., 125 , 7 (1997), 1903–191 2. [9] K. Kuratowski, T op olo gy. V ol. I. New e dition, r evise d and augm ente d , New Y ork-Lo ndo n: Academic Press ; W arsza w a: PWN-P olish Scientific Publishe rs. XX, 1966. [10] M. Rieffel, Gr oup C ∗ -algebr as a s c omp act quantum metric sp ac es , Do c. Math., 7 (2002), 605–651. [11] A. P apadop oulos, Metric sp ac es, c onvexity an d nonp ositive curvatur e , IRMA Lectures in Mathematics and Theoretical Phis ics, 6. EM S, 2005. [12] W.Rinow, Die inner e Ge ometrie der metrischen R¨ aume , Grund. Math. Wiss., vol. 10 5, Berlin-G¨ ottingen-Heidelb erg: Springer-V erlag, 1961. [13] C. W ebster, A. Winc hester, Boundaries of hyp erb olic metric sp ac es , Pac. J. Math. 221 , No. 1 (2005), 1 47-158. [14] C. W alsh, Hor ofunction b oundary of fin ite dimension a l norme d sp ac e // Pre prin t, ArXiv:math.MG/0510105. Pomor S tate University, Arkhangelsk, R uss ia Email: pdandreev@m ail.ru 28