math.GT 2008-01-15 0

The coarse classification of homogeneous ultra-metric spaces

We prove that two homogeneous ultra-metric spaces $X,Y$ are coarsely equivalent if and only if $\mathrm{Ent}^\sharp(X)=\mathrm{Ent}^\sharp(Y)$ where $\mathrm{Ent}^\sharp(X)$ is the so-called sharp entropy of $X$. This classification implies that each…

Authors: Taras Banakh, Ihor Zarichnyy

THE CO ARSE CLASSIFICA TION O F HOMOGENEO US UL TRA-METRIC SP ACES T ARAS BANAKH, IHOR ZARICHNYY Abstract. W e prov e that t wo homogeneo us ultra-metric spaces X, Y are coarsely equiv alent if and only if Ent ♯ ( X ) = En t ♯ ( Y ) where Ent ♯ ( X ) is the so- called sharp entrop y of X . This classification im pl ies that eac h homogeneous proper ultra-metric spac e is coarsely equiv alent to the anti-Can tor set 2 <ω . F or the pr oof of these results we dev elop a tec hnique of to w ers which can ha v e an independen t interest. Contents Int ro duction 1 1. Character iz ing the coarse equiv alence 4 2. T owers 5 3. Admissible morphisms of tow ers 10 4. Asymptotically homogeneous tow ers 12 5. Pro of of Theorem 2. 14 6. Pro of of Prop os itio n 1. 14 7. Pro of of Theorem 3. 16 8. Some Ope n P roblems 17 References 19 Introduction In this pap er we classify homoge neous ultra-metric spaces up to the coarse equiv- alence. Let us recall some necessary definitions. W e say that a metr ic spac e ( X , d ) is • homo gene ous if for any tw o points x, y ∈ X there is an isometrical bijection f : X → X with f ( x ) = y ; • pr op er if X is unbounded but for ev ery x 0 ∈ X and r ∈ [0 , + ∞ ) the clo sed r -ba ll B r ( x 0 ) = { x ∈ X : d ( x, x 0 ) ≤ r } centered at x 0 is compact; • an ultr a-metric sp ac e if d ( x, y ) ≤ max { d ( x, z ) , d ( z , y ) } for e very p o int s x, y , z ∈ X . The basic example of a homogeneous prop er ultra-metric space is the space 2 <ω = { ( x i ) i ∈ ω ∈ 2 ω : ∃ m ∈ ω ∀ i ≥ m x i = 0 } endow ed with the ultrametric D ( ~ x, ~ y ) = max n ∈ ω 2 n | x n − y n | , 1991 Mathematics Subje c t Classific ation. 54E35, 54E40 . 1 2 T ARAS BANAKH, IHOR ZARICHNYY where ~ x = ( x n ) n ∈ ω and ~ y = ( y n ) n ∈ ω are tw o p oints of 2 <ω . Here 2 = { 0 , 1 } and more generally , α = { β : β < α } for any o rdinal α . The ultra-metric space 2 <ω , called the anti-Cantor set, is an as ymptotic coun- terpart of the Cant or cube 2 ω endow ed with the ultrametric d ( ~ x, ~ y ) = max n ∈ ω 2 − n | x n − y n | By analog y , for ev ery set A with | A | > 1 w e can consider the co untable prod- uct ( A ω , d ) and its asymptotic coun terpart ( A <ω , D ). According to the classical Brouw er theorem for each finite set A with | A | > 1 the coun table pro duct A ω is (uniformly) homeomor phic to the Cant or cub e 2 ω . The problem if the Bro uw er theorem has a n asy mpto tic counterpart has be e n circulated among asymptolog ists (see [BDHM, § 5]) and w as comm unicated to the authors b y Ihor Pro tasov. T o answer this question we first need to reca ll the notion of the co arse equiv alence, which relies on the notion o f a bor nologo us map. By definition, a map f : X → Y betw een metr ic spaces is b ornolo gous if for every ε ∈ R there is δ ∈ R such that for each p oints x, x ′ ∈ X with dist ( x, x ′ ) ≤ ε w e get dist ( f ( x ) , f ( x ′ )) ≤ δ . Definition 1. W e say that t wo metric spaces X , Y are • bije ctively asymorphic if there is a b or nologous bijective map f : X → Y with bo rnologo us in v erse f − 1 ; • c o arsely e quivalent if ther e a re b orno logous maps f : X → Y and g : Y → X such tha t dist ( g ◦ f , id X ) < ∞ and dist ( f ◦ g , id Y ) < ∞ . In Section 1 we s hall give several equiv alent definitions of the coarse equiv alence. It is k nown that for tw o finite sets A, B the metric spaces A <ω and B <ω are bijectiv ely asymorphic if and o nly if | A | a nd | B | have the same prime divisors , see [PB, 10.6], [PZ, p.57 ] or [BDHM , 5.5 ]. In particular , 2 <ω and 3 <ω are not bijectiv ely asymorphic. In light of this result, it is natural to a sk if 2 <ω and 3 <ω are coa rse equiv ale nt, see [BDHM, § 5]. T he p o sitive answer to this question can be eas ily derived from the homogeneit y of 2 <ω and 3 <ω and the following theo r em (that is a particular case of a more general Theorem 2 below). Theorem 1. Any homo gene ous pr op er ultr a-metric sp ac e is c o arsely e quivalent to the ant i-Cantor set 2 <ω . According to [Roe, 2.42], an y tw o coars e ly equiv alent prop er metric spaces X , Y hav e ho meo morphic Higson coronas ν X , ν Y . Combining this fact with Theorem 1, we get Corollary 1. The Higson c or onas ν X , ν Y of any t wo homo gene ous pr op er ultr a- metric s p ac es X, Y ar e home omorphic. Theorem 1 follows from a more general re s ult detecting ultra-metric s paces coarsely equiv a lent t o the Ca ntor set with helf of cardinal inv ariants called small and lar ge en tropies. Given a subset B of a metr ic space X and a r eal num ber ε we define the ε -entr opy Ent ε ( B ) of B as the smalles t cardina lit y | N | of an ε - ne t N ⊂ B (the latter means tha t for each point x ∈ B there is a po int y ∈ N with dist ( x, y ) < ε ). F o r ε, δ ∈ [0 , ∞ ) let Ent δ ε ( X ) = sup x ∈ X Ent ε ( B δ ( x )) and en t δ ε ( X ) = min x ∈ X Ent ε ( B δ ( x )) THE CO ARSE CLASSIFICA TION OF HOM OGENEOUS UL TRA-METRIC SP A CES 3 where B δ ( x ) = { y ∈ X : dist ( x, y ) ≤ δ } stands for the clos ed δ -ball centered at x . A metric space X is defined to hav e b ounde d ge ometry if there is ε ∈ R such that En t δ ε ( X ) < ℵ 0 for all δ ∈ R . F or such spaces w e hav e the f ollowing theo rem implying Theorem 1. Theorem 2. A pr op er u ltr a-metric sp ac e X is c o arsely e quivalent to the anti- Cantor set pr ovide d t her e is an incr e asing u nb ounde d numb er se quenc e ~ r = { r n } n ∈ N such that Y n ∈ N Ent r n +1 r n ( X ) ent r n +1 r n ( X ) < + ∞ . Theorem 1 is the principal ing r edient in the coarse classification of homogeneous ultra-metric spac e s . Such spac es are c lassified with help of a car dina l inv ariant called the sharp entr opy . T o define this car dinal inv aria n t, for a metric space X and a real n um b e r ε let Ent ♯ ε ( X ) = sup δ< ∞  Ent δ ε ( X )  + and en t ♯ ε ( X ) = sup δ< ∞  ent δ ε ( X )  + be the lar ge and sm al l ε - entr opies of X (here by κ + we denote the succ e ssor cardinal to a cardinal κ ). The cardinal num b er s Ent ♯ ( X ) = min ε< ∞ Ent ♯ ε ( X ) and en t ♯ ( X ) = min ε< ∞ ent ♯ ε ( X ) are called the lar ge and smal l sharp entr opies of X , resp ectively . It is clear that ent ♯ ( X ) ≤ Ent ♯ ( X ) for an y metr ic space X . If X is homogeneous, then w e hav e the equality ent ♯ ( X ) = Ent ♯ ( X ) (b eca use E nt ε ( B δ ( x )) = Ent δ ε ( B δ ( y )) for all ε, δ and x, y ∈ X ). It follo ws that En t ♯ ( X ) ≤ ℵ 0 if and only if there is ε > 0 suc h that Ent δ ε ( X ) < ℵ 0 for all δ ∈ R , which means that X ha s b ounded g e ometry . Observe that the sharp en tropy distinguishes betw een the anti-Can tor set 2 <ω and the an ti-Baire space N <ω bec ause Ent ♯ (2 <ω ) = ℵ 0 while Ent ♯ ( N <ω ) = ℵ 1 . The following classification theorem (implying Theorem 1) is one of the ma in result of this paper . Theorem 3. Two homo gene ous ultr a-metric sp ac es ar e c o arsely e quival ent if and only if Ent ♯ ( X ) = Ent ♯ ( Y ) . The following propositio n completes Theo rem 3 and presents some elemen tary prop erties of the sharp entropies. Prop ositi o n 1. (1) If a metric sp ac e X is c o arsely e quivalent to a subsp ac e of a met r ic sp ac e Y , t hen Ent ♯ ( X ) ≤ Ent ♯ ( Y ) . (2) If two metric sp ac es X , Y ar e c o arsely e quivalent, then Ent ♯ ( X ) = Ent ♯ ( Y ) and en t ♯ ( X ) = ent ♯ ( Y ) . (3) An ultr a-metric sp ac e X is c o arsely e quival ent t o a su bsp ac e of an u ltr a- metric s p ac e Y pr ovide d Ent ♯ ( X ) ≤ ent ♯ ( Y ) . (4) F or a c ar dinal num b er κ the r e is a non-empty (pr op er h omo gene ous ultra -) metric sp ac e X with Ent ♯ ( X ) = κ if and only if either κ = 2 or κ is an infinite suc c essor c ar dinal, or κ is a limit c ar dinal of c ount able c ofinality. The third item of the preceding pro po sition generalizes a result of A.Dranishniko v and M.Zaric hnyi [DZ] who proved that eac h ultra -metric space X of b o unded ge- ometry is coarsely equiv a le nt to a subspace of the anti-Cantor set 2 <ω . 4 T ARAS BANAKH, IHOR ZARICHNYY In fact, the a b ov e r e sults apply not only to ( homogeneo us) ult ra-metr ic spaces but, more g e nerally to asymptotically zero-dimensio na l (homogeneous) metric spaces bec ause an y such a space is bijectively a symorphic to a (homogeneous) ultr a-metric space, see Prop ositio n 7. 1. Characterizing the coarse equiv al ence In this section we show that v ario us natur al ways of defining morphisms in Asymptology 1 lead to the same notion of coa rse equiv alence. B e sides the o r iginal approach of J. Ro e based on the notion of a coa rse map, w e discuss an alternative approach based on the notion of a multi-map. By a multi-map Φ : X ⇒ Y b etw een t wo s ets X , Y w e understand any subset Φ ⊂ X × Y . F o r a subset A ⊂ X by Φ( A ) = { y ∈ Y : ∃ a ∈ A with ( a, y ) ∈ Φ } we denote the image of A under the multi-map Φ. The in v erse Φ − 1 : Y ⇒ X to the m ulti-map Φ is the s ubs et Φ − 1 = { ( y , x ) ∈ Y × X : ( x, y ) ∈ Φ } ⊂ Y × X . F or tw o m ulti-maps Φ : X ⇒ Y and Ψ : Y ⇒ Z w e define their co mpo sition Ψ ◦ Φ : X ⇒ Z as usual: Ψ ◦ Φ = { ( x, z ) ∈ X × Z : ∃ y ∈ Y such that ( x, y ) ∈ Φ a nd ( y , z ) ∈ Ψ } . A multi-map Φ is called surje ctive if Φ( X ) = Y and bije ctive if Φ ⊂ X × Y coincides with the graph of a bijectiv e (single-v alued) function. A m ulti-map Φ : X ⇒ Y b etw een metric spaces X and Y is ca lled • b ornolo gous if for any ε > 0 there is δ > 0 such that for a ny subset A ⊂ X with diam ( A ) < ε the imag e B = Φ( A ) has diameter diam ( B ) < δ ; • an asymorphism if b oth Φ and Φ − 1 are s ur jective borno logous multi-maps; • an asymorphi c emb e dding if b oth Φ and Φ − 1 are bo rnolog ous multi-maps and Φ − 1 is surjective. It is clear that the comp os ition of t wo s ur jective (b orno logous) multi-maps is surjective (and b o rnologo us). Consequently , the co mpo sition of asymo r phisms is an asymorphism. Definition 2. W e sha ll s ay that tw o metric spaces X , Y are ( bije ctively ) asy- morphic 2 and will denote this by X ∼ Y if there is a (bijective) asymor phism Φ : X ⇒ Y . A subs e t L of a metric space X is called lar ge if O r ( L ) = X for some r ∈ R , where O r ( L ) = { x ∈ X : dist ( x, L ) < ε } stands for the open r -neighbo rho o d o f the set L in X . The following characterizatio n is the main (and unique) result of this section. Prop ositi o n 2. F or metric sp ac es X, Y the fol lowing assertions ar e e quivalent: (1) X and Y ar e asymorphic; (2) X and Y ar e c o arsely e quivalent; (3) the sp ac es X, Y c ont ain bije ctively asy morphic lar ge subsp ac es X ′ ⊂ X and Y ′ ⊂ Y ; 1 The term “Asymptology” was introduced b y I .Protasov i n [ PZ] for naming the the ory study ing large scale properties of metric spaces (or more general ob jects like b al lea ns of I. Protaso v [ PZ], [PB] or c o arse structur es of J. Ro e [Ro e]). 2 In [PZ] bij ectiv e asymorphisms are called asymorphisms while asymorphisms are referred to as quasi-asymorphisms. H ow ev er w e suggest to change the terminology shifting the att en tion to asymorphisms (in our sense) as a cen tral concept of the Asymptology . THE CO ARSE CLASSIFICA TION OF HOM OGENEOUS UL TRA-METRIC SP A CES 5 (4) ther e ar e two b ornolo gous maps f : X → Y , g : Y → X whose inverses f − 1 : Y ⇒ X and g − 1 : X ⇒ Y ar e b ornolo gous multi-maps and max { dist ( g ◦ f , id X ) , dist ( f ◦ g , id Y ) } < ∞ . Pr o of. T o pr ov e the equiv alence o f the items (1)– (4), it suffices to establish the implications (1) ⇒ (4) ⇒ (2) ⇒ (3) ⇒ (1). (1) ⇒ (4 ) Assuming that X and Y are asymorphic, fix a surjective borno logous m ulti-map Φ : X ⇒ Y with surjective b ornolog ous in v erse Φ − 1 : Y ⇒ X . Since the m ulti-map Φ − 1 is surjective, for every x ∈ X there is a po int f ( x ) ∈ Y with x ∈ Φ − 1 ( f ( x )), which is equiv alent to f ( x ) ∈ Φ( x ) . It follows from the bo rnolog ity of Φ that the map f : X → Y is b orno lo gous. Since f − 1 ( y ) ⊂ Φ − 1 ( y ) for all y ∈ Y , the bo rnolog ous pr op erty of Φ − 1 implies that prop erty for the m ulti-ma p f − 1 : Y ⇒ X . By the same r e a son, the surjectivit y of the multi-map Φ implies the existence of a map g : Y → X such that g ( y ) ∈ Φ − 1 ( y ) for all y ∈ Y . The bo r nologity of Φ and Φ − 1 implies that g : Y → X a nd g − 1 : X ⇒ Y a re borno logous. Since the comp o s ition Φ − 1 ◦ Φ : X ⇒ X is b ornolog ous, ther e is a constant C < ∞ such that diam Φ − 1 ◦ Φ( x ) ≤ C . Observing that { x, g ◦ f ( x ) } ⊂ Φ − 1 ◦ Φ( x ) we see that dist ( g ◦ f , id X ) ≤ C < ∞ . By t he same reaso n, dist ( f ◦ g , id Y ) < ∞ . The implication (4) ⇒ (2 ) trivially fo llows fro m the definition of the coarse equiv ale nc e given in the Intro duction. (2) ⇒ (3) Assume tha t there are t wo borno logous maps f : X → Y , g : Y → X with dist ( g ◦ f , id X ) ≤ R and dist ( f ◦ g , id Y ) ≤ R for so me real num b er R . It fo llows that O R ( f ( X )) = Y a nd hence the set Y ′ = f ( X ) is la r ge in Y . Cho ose any subset X ′ ⊂ X ma king the restrictio n h = f | X ′ : X ′ → Y ′ bijectiv e. The bo rnolog ous prop erty of f implies that the bijective map h : X ′ → Y ′ is b ornolo g ous. Let us show that the inv erse map h − 1 : Y ′ → X ′ is b or nologo us . Given ar bitrary ε < ∞ , use the bornolog ity of the map g : Y → X to find a num b e r δ < ∞ such that d iam g ( C ) < δ for ev ery set C ⊂ Y with dia m ( C ) ≤ ε . Now take any p oints y , y ′ ∈ Y ′ with dist ( y , y ′ ) ≤ ε and let x = h − 1 ( y ) and x ′ = h − 1 ( y ′ ). W e claim that dist ( x, x ′ ) ≤ δ + 2 R . Indeed, t he choice of δ guar antees that dist ( g ( y ) , g ( y ′ )) ≤ δ . Since dist ( g ◦ f , id X ) ≤ R , w e conclude that dist ( x, x ′ ) ≤ dist ( x, g ◦ f ( x )) + dist ( g ◦ f ( x ) , g ◦ f ( x ′ )) + dist ( g ◦ f ( x ′ ) , x ′ ) ≤ ≤ R + dist ( g ( y ) , g ( y ′ )) + R ≤ δ + 2 R. Finally , let us sho w that the set X ′ is lar ge in X . Given any point x ∈ X , find a p oint x ′ ∈ X ′ with f ( x ) = f ( x ′ ). Then dist ( x, x ′ ) ≤ dist ( x, g ◦ f ( x )) + dist ( g ◦ f ( x ′ ) , x ′ ) ≤ 2 R a nd consequently , O 2 R ( X ′ ) = X . (3) ⇒ (1) Assume that the spaces X, Y contain bijectively as ymorphic la rge subspaces X ′ ⊂ X and Y ′ ⊂ Y . Let f : X ′ → Y ′ be a bijectiv e asymor phism. Find R ∈ R such tha t O R ( X ′ ) = X and O R ( Y ′ ) = Y . T ake any ma ps ϕ : X → X ′ and ψ : Y → Y ′ with dist ( ϕ, id X ) ≤ R and dist ( ψ , id Y ) ≤ R . It is ea sy to see that ϕ a nd ψ ar e as ymorphisms and then the comp ositio n ψ − 1 ◦ f ◦ ϕ : X ⇒ Y is a required asymorphism betw een X and Y .  2. Towe rs The results stated in the In tro duction are pr ov ed by inductio n on partially o r- dered s ets called tow ers. T ow ers are order a ntipo des o f trees but on the other hand, 6 T ARAS BANAKH, IHOR ZARICHNYY seen as graphs, the to wers are trees in the graph-theo r etic s e nse (i.e., are connected graphs without circuits). W e r ecall that a par tia lly o rdered set T is a tr e e if T has the s mallest elemen t and for ev ery point x ∈ T the low er cone ↓ x is well-ordered. By the lower c one (r e sp. upp er c one ) of a point x of a par tia lly or dered set T we understand the set ↓ x = { y ∈ T : y ≤ x } (resp. ↑ x = { y ∈ T : y ≥ x } ). A subset A ⊂ T will b e ca lled a lower (res p. upp er ) set if ↓ a ⊂ A (r esp. ↑ a ⊂ A ) for all a ∈ A . A partially o rdered set T is wel l-founde d if each subset A ⊂ T has a minimal element a ∈ A . The minimalit y of a means th at eac h point a ′ ∈ A with a ′ ≤ a is equal to a . By min T we shall deno te the set of all minimal elemen ts of T . Now we define the pr incipal technical c oncept of this pap er. Definition 3. A partially order e d set T is called a tower if (1) T is well-founded; (2) any tw o elemen ts x, y ∈ T hav e the smalle s t upper b ound s up( x, y ) in T ; (3) for any x ∈ T the upp er cone ↑ x is linear ly ordered; (4) for any p oint a ∈ T there is a finite num b er n = lev T ( a ) such that for every minimal elemen t x ∈ ↓ a of T the order interv al [ x, a ] = ↑ x ∩ ↓ a has cardinality   [ x, y ]   = n . The function lev T : T → N , lev T : a 7→ lev T ( a ), fr o m the last item is called the level function . If the tow er T is clear from the context, then we omit the subscript T and write lev ( a ) instea d of lev T ( a ). One can o bserve that lev T = 1 + rank T where rank T is the usual rank function of the well-founded set T , see [Ke , App e ndix B]. The lev el function lev T : T → N divides T into the levels L i = lev − 1 T ( i ), i ∈ N . The 1- s t level L 1 = min T will be called the b ase of T and will be denoted b y [ T ]. The n um ber h ( T ) = sup { n ∈ N : L n 6 = ∅ } is called the heigh t o f the tow er T . A tow er T is un b ounde d if it has infinite height. The following model of the famo us Eiffel to wer is just an example of a tow er of heig ht 7 . T [ T ] r r r r r ✓ ✓ ❙ ❙ r   r ❅ ❅ r ❈ ❈ ✄ ✄ r r r ✲ lev T ✻ r 7 r 6 r 5 r 4 r 3 r 2 r 1 In fact, t ow ers of finite heig h t are not in teresting: they are trees in the reverse partial order. Because of that we sha ll assume that all to wers are unbounded. Each tower carr ie s a ca nonic p ath metric d T defined by the formula d T ( x, y ) = 2 · lev T  sup( x, y )  −  lev T ( x ) + lev T ( y )  for x, y ∈ T . The path metr ic d T restricted to the base [ T ] of T is an ultrametric. In the sequel talking ab out metric prop erties of tow ers we shall alw ays refer to the path metric. A subs e t S of an tow er T is c alled a subtower if S is an tow er in the induced partial or der. F or ev ery tow er T and an incre a sing num ber sequence ~ k = ( k n ) n ∈ ω THE CO ARSE CLASSIFICA TION OF HOM OGENEOUS UL TRA-METRIC SP A CES 7 the subset T ( ~ k ) = { x ∈ T : lev( x ) ∈ { k n } n ∈ ω } is a subtow er of T , called the level subtower o f T generated b y the sequence ~ k , or briefly the level ~ k - subtower of T . It is e asy to see that ea ch un b o unded subtow er S of a tow er T is c ofinal in T in the sense that fo r ev ery t ∈ T ther e is s ∈ S with t ≤ s . Given a co final subset S ⊂ T consider the map next S : T → S a ssigning to each x ∈ T the smallest po int y ∈ S w ith y ≥ x (suc h a smallest point exists because the upp er set ↑ x is well-ordered). It is ea sy to see that next S ([ T ]) ⊂ [ S ]. The following pro p osition trivially follows from the definitions. Prop ositi o n 3. L et T b e an t ower and S = T ( ~ k ) b e a lev el subtower of T . Then the m ap next S : [ T ] → [ S ] is an asymo rphism. F or every p oint x ∈ T of a tow er T and a num ber i ≤ lev( x ) let pred i ( x ) = L i ∩ ↓ x be the set of predecessors of x in the i -th generation a nd deg i ( x ) = | pr e d i ( x ) | . F or i = lev( x ) − 1 the s e t pred i ( x ) is called the set o f parents of x and is denoted by pred( x ) . The cardinality | pred( x ) | is ca lled the de gr e e of x and is denoted by deg( x ). Thus deg( x ) = deg lev( x ) − 1 ( x ). F or an integer n um ber s k ≤ n let deg n k ( T ) = min { deg k ( x ) : x ∈ L n } and Deg n k ( T ) = sup { deg k ( x ) : x ∈ L n } . W e shall write deg n ( T ) and Deg n ( T ) instead of deg n +1 n ( T ) and Deg n +1 n ( T ), resp ec- tively . The small and la rge entropies of the bo undary [ T ] of a tow er T can b e easily calculated via the degrees deg j i ( T ) and Deg j i ( T ) of T . Prop ositi o n 4. F or any tower T we h ave (1) ent 2 j 2 i ([ T ]) = deg j +1 i +1 ( T ) a nd En t 2 j 2 i ([ T ]) = Deg j +1 i +1 ( T )) + for a l l i ≤ j ; (2) ent ♯ ([ T ]) = min i ∈ N sup j >i (deg j i ( T )) + and E nt ♯ ([ T ]) = min i ∈ N sup j >i (Deg j i ( T )) + . This prop osition can be easily derived from the definition of the pa th metric o n the b oundary [ T ] of T a nd the definition of the small and large sharp entropies of [ T ]. In order to prov e a tow er coun terpart of Prop ositio n 1(3) we need a definition. An injeciv e (resp. bijective) map ϕ : T 1 → T 2 will be called a tower emb e dding (resp. a tow er i somorphism ) if • ϕ is monotone in the sense that x ≤ y in T 1 implies ϕ ( x ) ≤ ϕ ( y ) in T 2 and • level-pr eserving , whic h mea ns that lev T 2 ( ϕ ( x )) = le v T 1 ( x ) for all x ∈ T 1 . This definition com bined with the definition o f the path metric o f a tower implies Prop ositi o n 5. F or e ach tower emb e dding (isomorphism) ϕ : T 1 → T 2 the r est ric- tion ϕ | [ T 1 ] : [ T 1 ] → [ T 2 ] i s a n isometric emb e dding (bi je ction). Now we give c o nditions o f tow ers T 1 , T 2 guarantees the existence of a tow er embedding (isomorphism) T 1 → T 2 . Prop ositi o n 6. F or two towers T 1 , T 2 ther e is a t ower emb e dding (isomorph ism) ϕ : T 1 → T 2 pr ovide d Deg k ( T 1 ) ≤ deg k ( T 2 ) (and Deg k ( T 2 ) ≤ deg k ( T 1 ) ) for al l k ∈ N . 8 T ARAS BANAKH, IHOR ZARICHNYY Pr o of. Assume that Deg k ( T 1 ) ≤ deg k ( T 2 )  and Deg k ( T 2 ) ≤ deg k ( T 1 )  for all k ∈ N . W e s hall need the following Lemma 1. F or any two p oints u ∈ T 1 and v ∈ T 2 with lev( u ) = lev( v ) ther e is a tower emb e dding (isomo rphism) ϕ : ↓ u → ↓ v . Mor e over, if for some u 0 ∈ pred( u ) and v 0 ∈ pred( v ) we ar e given with a tower emb e dding (isomorphism) ϕ 0 : ↓ u 0 → ↓ v 0 , t hen the map ϕ c an b e chosen so t hat ϕ |↓ u 0 = ϕ 0 . Pr o of. The pro o f is b y induction of the le vel lev( u ) = lev( v ). If this level is 1, then there is nothing to construct: just put ϕ : { u } → { v } be the c onstant map. Now assume that the lemma ha s b een proved f or all u , v with lev ( u ) = lev( v ) < n . T ake any points u ∈ T 1 and v ∈ T 2 with lev ( u ) = lev ( v ) = n . Consider the sets pred( u ) and pred( v ). Since Deg n − 1 ( T 1 ) ≤ deg n − 1 ( T 2 ), we conclude that | pr e d( u ) | ≤ | pred( v ) | a nd thus we ca n constr uct a n injective ma p ξ : pred( u ) → pre d( v ). If Deg n − 1 ( T 2 ) ≤ deg n − 1 ( T 1 ), then pred( u ) | = | pred( v ) | and we can take ξ to b e bijectiv e. F or ev ery u ′ ∈ pred( u ) use the inductive assumption to find a tower embedding (isomorphism) ϕ u ′ : ↓ u ′ → ↓ ξ ( u ′ ). T he maps ϕ u ′ , u ′ ∈ pred( u ), can be unified to comp ose a tow er em bedding ϕ : ↓ u → ↓ v such that ϕ ( u ) = v and ϕ ( x ) = ϕ u ′ ( x ) for each x ∈ ↓ u ′ with u ′ ∈ pred( u ). If for so me u 0 ∈ pred( u ) and v 0 ∈ pred( v ) w e had a to wer embedding (isomor- phism) ϕ 0 : ↓ u 0 → ↓ v 0 , then we can c ho ose the injection ξ so that ξ ( u 0 ) = v 0 and take ϕ u 0 be equal to ϕ 0 .  Now the pr o of of Pr op osition 2 b ecomes ea sy . Fix a ny tw o po int s x 1 ∈ [ T 1 ] and y 1 ∈ [ T 2 ] and consider the upper cones ↑ x 1 = { x k : k < h ( T 1 ) + 1 } and ↑ y 1 = { y k : k < h ( T 2 ) + 1 } where lev( x k ) = k = lev( y k ) for all k . Using Lemma 1, constr uc t a sequence of tow er e m b eddings (isomorphisms) ϕ n : ↓ x n → ↓ y n such that ϕ n +1 |↓ x n = ϕ n for all n < h ( T 1 ) + 1. Unifying these em- bedding s w e obtain a desired tow er em bedding (iso morphism) ϕ : T 1 → T 2 defined by ϕ ( x ) = ϕ n ( x ) for x ∈ ↓ x n .  W e define a to w er T to be homo gene ous if deg n ( T ) = Deg n ( T ) for a ll n ∈ N (and consequently , deg n k ( T ) = Deg n k ( T ) for all k ≤ n ). Applying Prop o sition 6 to homogeneous tow ers w e get Corollary 2. F or two homo gene ous towers T 1 , T 2 ther e is a t ower isomorphism ϕ : T 1 → T 2 if a nd o nly if deg k ( T 1 ) = deg k ( T 2 ) for al l k ∈ N . A typical example of a homogeneous tow er can b e constructed as follows. Let G be a group written as the countable union G = S n ∈ N H n of a n increas ing sequence of subgroups H n ⊂ H n +1 . The set T = { g H n : g ∈ G, n ∈ N } is an tow er with resp ect to the inclusion order ( A ≤ B iff A ⊂ B ). Observe that the degree of an y element g H n in T is equal to the index of the subgr oup H n − 1 in the gro up H n (here we assume that the subgr oup H 0 is trivial). In pa rticular, for every sequence ~ k = ( k n ) n ∈ N of p ositive integers we can con- sider the direct sum G = ⊕ n ∈ N Z /k n Z of cyclic gr oups a nd the subg r oups H n = ⊕ i 0, then the asymorphness of X a nd [ T X ( ~ r )] follows fro m Prop osition 3.  It is known (see Theorems 3.1 .1 and 3 .1.3 in [PZ ]) that a metric s pace X is bijectiv ely asymorphic to an ultra-metric space if a nd only if X is asymptotica lly zero-dimensio nal. The latter means that for every real num ber D > 0 there is a D -discrete cov er U of X with mesh ( U ) = sup U ∈U diam U < + ∞ . The D -discr eteness of U means that dist ( U, V ) > D for any distinct sets U, V ∈ U . The followin g prop osition is a “homogeneo us” version of the men tioned result. Prop ositi o n 8. Each ( homo gene ous) asymptotic al ly zer o-dimensional metric sp ac e ( X, d ) admits an ultra metric ρ su ch that the met ric sp ac es ( X , d ) and ( X , ρ ) ar e bije ctively asymorphi c (and t he u lt r a-metric sp ac e ( X , ρ ) is homo gene ous) . Pr o of. Using the definition o f the asymptotic ze r o-dimensiona lit y o f X , construct an increasing seq uence ( r n ) n ∈ N of p ositive real n um b ers such that for every n ∈ N the space X has a r n -discrete cov er U n with mesh U n < r n +1 . Define tw o po int s x, y ∈ X to b e r n -equiv a lent if there is a chain o f p oints x = x 0 , x 1 , . . . , x k = y in X with dist ( x i − 1 , x i ) ≤ r n for a ll i ≤ k . It is clear that the r n -equiv a lence is indeed an equiv alence relation, whic h divides the space X in to the equiv alence classes. Let C x denote the equiv alence cla ss of a point x ∈ X a nd 10 T ARAS BANAKH, IHOR ZARICHNYY let C n = { C x : x ∈ X } . It is clear that the co ver C n is r n -discrete and B ( r n ) ≺ C n ≺ B ( r n +1 ) where B ( r ) = { B r ( x ) : x ∈ X } is the cover of X by clo sed r -balls, and fo r tw o cov ers U , V of X we write U ≺ V if each set U ∈ U lies in some set V ∈ V . Now define the ultra- metric ρ on X letting ρ ( x, y ) = ma x { n ∈ ω : { x, y } 6≺ C n } for differen t points x, y ∈ X . It is easy to see that the iden tit y map ( X, d ) → ( X, ρ ) is a bijective asymorphism and ea ch bijective isometry f : X → X of the metric space ( X , d ) is an isometry of the metric space ( X , ρ ). Consequen tly , the ultra - metric space ( X, ρ ) is homog eneous if so is the space ( X, d ).  3. Admissible morphisms of to wers Let T 1 , T 2 be t w o to wers. A m ap ϕ : A → T 2 defined on a lo w er subset A = ↓ A of T 1 is called an admissible morphism if (1) lev( ϕ ( a )) = lev( a ) for all a ∈ A ; (2) a ≤ a ′ in A implies ϕ ( a ) ≤ ϕ ( a ′ ); (3) ϕ ( a ) = ϕ ( a ′ ) for a, a ′ ∈ A implies t hat a, a ′ ∈ pr ed( v ) for some v ∈ T ; (4) ϕ ( A ) is a low er subse t of T 2 ; (5) | ϕ (max A ) | ≤ 1 , where max A stands for the (poss ibly empty) set of maxima l elements of the do main A . Lemma 2. L et ϕ : T 1 → T 2 b e an admissible morphism b etwe en towers T 1 , T 2 . Then the r estriction Φ = ϕ | [ T 1 ] : [ T 1 ] → [ T 2 ] i s a n asymo rphism. Pr o of. Given a ny n ∈ ω a nd a ny subset A ⊂ [ T 1 ] with diam A ≤ 2 n w e conclude that A ⊂ ↓ v for some v ∈ L n +1 . The monotonicity of ϕ implies that ϕ ( A ) ⊂ ϕ ( ↓ v ) = ↓ ϕ ( v ) and th us diam ( ϕ ( A )) ≤ diam ( ↓ ϕ ( v )) ≤ 2 n bec ause lev( ϕ ( v )) = lev( v ) = n + 1 . Now assume conv ersely that B ⊂ [ T 2 ] is a subset with diam ( B ) ≤ 2 n . W e cla im that diam ( ϕ − 1 ( B )) ≤ 2 n + 2. T ake any t wo p oints x, y ∈ ϕ − 1 ( B ). The inequality diam ( B ) ≤ 2 n implies that B ⊂ ↓ b for some b ∈ T 2 with lev( b ) = n + 1. Let x ′ , y ′ ∈ L n be tw o p oints with x ≤ x ′ and y ≤ y ′ . It follows that lev ( ϕ ( x ′ )) = lev( x ′ ) = n + 1 = le v ( y ′ ) = lev ( ϕ ( y ′ )). W e c laim that ϕ ( x ′ ) = b . F or the sma lle st low er bo und v = sup( b, ϕ ( x ′ )), consider the lower co ne ↓ v that con tains the point ϕ ( x ) as a minimal elemen t. Since the order in terv a l [ ϕ ( x ) , v ] is well-ordered and contains tw o elemen ts b and ϕ ( x ′ ) at the sa me level, we conclude that ϕ ( x ′ ) = b . By the sa me r eason ϕ ( y ′ ) = b . Since ϕ is an admissible morphism, the equalit y ϕ ( x ′ ) = ϕ ( y ′ ) implies that x ′ , y ′ ∈ pr ed( w ) for so me point w ∈ T . It follows that lev( w ) = lev( x ′ ) + 1 = lev( b ) + 1 = n + 2 and hence dist ( x, y ) = 2 lev(sup( x, y )) − 2 ≤ 2lev(sup( x ′ , y ′ )) − 2 ≤ 2 lev ( w ) − 2 = 2 n + 2 .  THE CO ARSE CLASSIFICA TION OF HOM OGENEOUS UL TRA-METRIC SP A CES 11 F or a rea l n um ber r denote b y ⌊ r ⌋ = min { n ∈ Z : r ≤ n } and ⌈ r ⌉ = max { n ∈ Z : r ≥ n } t wo nearest integer n um ber s to r . The following lemma is a cr ucial step in the proo f of T heo rem 2. Lemma 3. F or t wo towers T 1 , T 2 ther e is a su r je ctive admissible morphism ϕ : T 1 → T 2 pr ovide d ther e ar e two se quenc es ( a i ) i ∈ N and ( b i ) i ∈ N of re als such t hat 1 ≤ a i ≤ a i + 2 ≤ b i , ⌈ a i ⌉ ≤ deg i ( T 1 ) , and b i + a i · Deg i ( T 2 ) a i +1 ≤ deg i ( T 1 ) ≤ Deg i ( T 1 ) ≤ a i + b i ·  deg i ( T 2 ) b i +1 − 2  for al l i ∈ N . Pr o of. W e define a subset A ⊂ T 1 to b e admi ssible if A ⊂ pred( v ) for some v ∈ L k , k ∈ ω , and a k ≤ | A | ≤ b k . In this case we write v = sup( A ). Our lemma will be der ived from the fo llowing Claim 1. F or any admissible subset A ⊂ T 1 and any w ∈ T 2 with lev ( A ) = lev ( w ) ther e is an admissible morphism ϕ : ↓ A → ↓ w ⊂ T 2 . Mor e over, if we had an admissible morphism ϕ 0 : ↓ A 0 → ↓ w defin e d on the lower set of an admissible subset A 0 ⊂ ↓ A with sup A 0 ∈ A , then the admissible m orphism ϕ c an b e chosen so t hat ϕ |↓ A 0 = ϕ 0 . This claim will be prov en by induction on the level lev( w ) of the point w ∈ T 2 . If lev ( w ) = 1, then ther e is no ting to construct: j ust take ϕ : ↓ A → { w } b e the co nstant map. Assume that the claim is prov ed for all p oints w ∈ T 2 with lev( w ) ≤ n . T ake any point w ∈ T 2 with lev ( w ) = n + 1 and let A ⊂ T 1 be an admissible subset with lev ( A ) = lev( w ) = n + 1. F or every p o int x ∈ A choos e a num b er d x ∈ { ⌊ deg( w ) / | A |⌋ , ⌈ deg ( w ) / | A |⌉} so that P x ∈ A d x = deg ( w ). F or every x ∈ A write the set pred( x ) as a disjoint union pred( x ) = ∪A x of a family of admissible sets with cardinality |A x | = d x . This is p ossible b ecause b n + a n ( d x − 1) ≤ b n + a n deg( w ) | A | ≤ b n + a n Deg n ( T 2 ) a n +1 ≤ ≤ deg n ( T 1 ) ≤ deg ( x ) ≤ Deg n ( T 1 ) ≤ a n + b n ·  deg n ( T 2 ) b n +1 − 2  ≤ ≤ a n + b n ·  deg( w ) | A | − 2) ≤ a n + b n · ( d x − 1) . Moreov er, those inequalities guara nt ee tha t we can choose the family A x to con tain an admissible set of an y cardinality b etw een a n and b n . Then t he family A = S x ∈ A A x has ca rdinality |A| = deg( w ) and h ence w e ca n find a bijective map f : A → pred( w ). By the inductiv e assumption, f or each set A ′ ∈ A we can find an a dmiss ible surjective homomo rphism ϕ A ′ : ↓ A ′ → ↓ f ( A ′ ). Now define an admissible homomor phism ϕ : ↓ A → ↓ w letting ϕ ( x ) = ( ϕ A ′ ( x ) if x ∈ ↓ A ′ for some A ′ ∈ A ; b if x ∈ A. If for some admissible subset A 0 ⊂ ↓ A with sup A 0 ∈ A we are given with an admissible morphism ϕ 0 : ↓ A 0 → ↓ w , then we can include the admissible set A 0 int o 12 T ARAS BANAKH, IHOR ZARICHNYY the family A and c hoo se the admissible morphism ϕ A 0 equal to ϕ 0 . This co mpletes the pro of of Claim 1. T o prove the lemma, take increas ing sequences { x n : n ∈ N } ⊂ T 1 and { y n : n ∈ N } ⊂ T 2 with lev( x n ) = n = lev( y n ) for a ll n ∈ N . F or ev ery n ∈ N b y induction choose an a dmissible subset A n ⊂ T 1 such that x n ∈ A n ⊂ pred( x n +1 ). Suc h a choice is p os sible b ecause ⌈ a n ⌉ ≤ deg n ( T 1 ) ≤ deg ( x n +1 ). Then ↓ A n ⊂ ↓ A n +1 . Using Cla im 1, we can construct a sequence ϕ n : ↓ A n → ↓ y n , n ∈ N , of surjectiv e admissible morphisms suc h that ϕ n +1 |↓ A n = ϕ n . The union ϕ = S n ∈ N ϕ n : T 1 → T 2 is a well-defined admiss ible morphis m.  4. Asymptoticall y homogeneous towers In this section we shall apply Le mma 3 in or der to prove that the base [ T ] of each asymptotica lly homog eneous tow er T is asymor phic to the anti-Cantor set. Let us obs erve that a to wer T is proper (as a metric space) if [ T ] is unbounded in the path metric of T and the low er se t ↓ x of e a ch p oint x ∈ T is finite. Definition 4. A tow er T is ca lled asymptotic al ly homo gene ous if T is pr op er and there is a real constant C such that m Y k = n Deg k ( T ) deg k ( T ) ≤ C for every k ≤ n . This is equiv alent to saying tha t the infinite pro duct ∞ Y k =1 Deg k ( T ) deg k ( T ) is conv ergent . The following lemma is a cr ucial step in the proo f of T heo rem 4 below. Lemma 4. F or any asymptotic al ly homo gene ous tower T ther e ar e r e al se quenc es ( a n ) , ( b n ) , and incr e asing nu mb er se quen c es ( n i ) , and ( m i ) su ch t hat (1) 1 ≤ a i ≤ a i + 2 ≤ b i , a i + 1 ≤ deg n i +1 n i ( T ) and (2) b i + a i 2 m i +1 − m i a i +1 ≤ deg n i +1 n i ( T ) ≤ Deg n i +1 n i ( T ) ≤ a i + b i  2 m i +1 − m i b i +1 − 2  for al l i ∈ N . Pr o of. Those sequences will b e constructed by inductio n. How e ver we s hould first make so me preparatory work. The asymptotic homogeneit y of T allows us to find a sequence of real n um be rs c i > 1 , i ∈ N , such that Deg i ( T ) ≤ c i · deg i ( T ) , i ∈ N , and the infinit e product Q ∞ i =1 c i conv erges to some real n um b er C ∞ 1 . Also fix an y sequence of real n umbers δ i > 1 with con v ergent infinite pro duct Q ∞ i =1 δ i . F or num ber s 1 ≤ i ≤ j ≤ ∞ let C j i = j − 1 Y k = i c k and δ j i = j − 1 Y k = i δ k . THE CO ARSE CLASSIFICA TION OF HOM OGENEOUS UL TRA-METRIC SP A CES 13 T o simplify the notation, for i ≤ j we put d j i = deg j i ( T ) and D j i = Deg j i ( T ). It follows from the choice o f the n umbers c i that (3) D j i ≤ C j i · d j i . By induction, for every i ∈ N w e s hall construct rea l num bers a i , b i and p o sitive int egers n i , m i that satisfy the conditions (2) and (4) b i a i ≥ C ∞ i · δ ∞ i T o start the induction, le t n 1 = 1, m 1 = 0, and c hoo se any rea l num bers a 1 , b 1 satisfying the inequalities 1 ≤ a 1 < a 1 + 2 ≤ b 1 and b 1 ≥ a 1 · C ∞ 1 · δ ∞ 1 . Assume that the num bers a i , b i , n i , m i satisfying (2) and (4) hav e b een con- structed. Since the bas e [ T ] of T is unbounded, the s equence ( D n n i ) n ≥ n i is un b ounded. This fact combined with the a lmost homogeneity of T implies tha t the sequence ( d n n i ) n ≥ n i is unbounded to o . Co nsequently , there is a n um ber n i +1 > n i such tha t d n i +1 n i > ⌈ a i ⌉ and (5) d n i +1 n i − b i d n i +1 n i + 2 b i ≥ 1 δ n i . Next, find a n um b er m i +1 > m i such that (6) 2 m i +1 − m i ( C ∞ n i +1 · δ ∞ n i +1 − 1) a i d n i +1 n i − b i > 2 and the n um b ers a i +1 and b i +1 defined by (7) a i +1 = 2 m i +1 − m i a i d n i +1 n i − b i and b i +1 = 2 m i +1 − m i b i C n i +1 n i d n i +1 n i + 2 b i − a i are greater than 1. W e claim that the so defined n um be rs n i +1 , m i +1 , a i +1 , b i +1 , satisfy the inductiv e assumptions. In fact, the condition (2) follo ws dir ectly from the definitions of the num bers a i +1 and b i +1 and the inequality (3). T o see that (4) also holds, observe that b i +1 a i +1 = b i a i · d n i +1 n i − b i C n i +1 n i d n i +1 n i + 2 b i − a i ≥ b i a i · 1 C n i +1 n i · d n i +1 n i − b i d n i +1 n i + 2 b i ≥ and us ing (5), the tr iv ial inequality δ n i +1 n i ≥ δ n i , and the inductive assumption b i a i ≥ C ∞ n i · δ ∞ n i we can co nt inue as ≥ b i a i · 1 C n i +1 n i · 1 δ n i ≥ C ∞ n i · δ ∞ n i · 1 C n i +1 n i · 1 δ n i +1 n i = C ∞ n i +1 · δ ∞ n i +1 . The lo wer b ound b i +1 a i +1 ≥ C ∞ n i +1 · δ ∞ n i +1 > 1 com bined with the ch oice of m i +1 in (6) yields the condition (1): b i +1 − a i +1 =  b i +1 a i +1 − 1  a i +1 ≥ ( C ∞ n i +1 · δ ∞ n i +1 − 1) 2 m i +1 − m i a i d n i +1 n i − b i > 2 . This finishes the inductive step, and als o the proo f of the lemma.  W e apply Lemm as 3 and 4 to prov e the main result of this section. 14 T ARAS BANAKH, IHOR ZARICHNYY Theorem 4. The b ase [ T ] of e ach asymptotic al ly h omo gene ous tower T is asymor- phic t he anti-Cantor set 2 <ω . Pr o of. Let ( a i ) ∞ i =1 , ( b i ) ∞ i =1 , ~ n = ( n i ) ∞ i =1 and ~ m = ( m i ) ∞ i =1 be the se q uences fro m Lemma 4. Let T ( ~ n ) b e the level ~ n -subtow er of T . By P rop osition 3, the map next 1 : [ T ] → [ T ( ~ n )] is an asymor phism. By the sa me rea son, the map next 2 : [ T 2 ] → [ T 2 ( ~ m )] fro m the ba s e of the binary tow er T 2 to the base of its level ~ m - subtow er T 2 ( ~ m ) is an asymorphism. Observe that deg i ( T ~ 2 ( ~ m )) = 2 m i +1 − m i , deg i ( T ( ~ n )) = deg n i +1 n i ( T ) , Deg i ( T ( ~ n )) = Deg n i +1 n i ( T ) which allows us to apply Le mma 3 to find a n admissible mo rphism ϕ : T ( ~ n ) → T 2 ( ~ m ). By Lemma 2, ϕ induces an asy mo rphism b etw een the bases [ T ( ~ n )] and [ T 2 ( ~ m )]. Finally we obtain an asymor phism betw een [ T ] and the anti-Cantor set 2 <ω as the comp osition of the asymorphisms [ T ] ∼ [ T ( ~ k )] ∼ [ T 2 ( ~ m )] ∼ [ T 2 ] ∼ 2 <ω .  5. Proof of Theorem 2. W e should prove that an un bounded ultra-metric s pa ce X of bo unded geometry is asymor phic to the anti-Canor set provided there is an incr easing unbounded sequence ~ r = ( r n ) n ∈ N such that (8) Y n ∈ N Ent r n +1 r n ( X ) ent r n +1 r n ( X ) < ∞ . By P r op osition 7, X is asy morphic to the base [ T X ( ~ r )] o f the canonic ~ r -tow er T X ( ~ r ) = { ( B r n ( x ) , n ) : x ∈ X , n ∈ N } . The entropy co ndition (8) is equiv ale nt to the asymptotic ho mogeneity of the to w er T X ( ~ r ). Applying Theo rem 4 , we conclude that the ant i-Cantor set 2 <ω is a symorphic to the base [ T X ( ~ r )] of T X ( ~ r ) and hence is also asymorphic to X . 6. Proof of P roposition 1 . 1. Ass ume tha t a metric space X is coars ely equiv a lent to a subs pace Z of a metric space Y . By Prop os ition 2, ther e is an asymorphism Φ : X ⇒ Z ⊂ Y , which is an asymorphic embedding of X into Y . Find ε > 0 such that Ent ♯ ε ( Y ) = E nt ♯ ( Y ). Since Φ − 1 : Y ⇒ X is borno lo gous, there is ε ′ > 0 suc h that diam (Φ − 1 ( B )) ≤ ε ′ for every bounded subset B ⊂ Y with diam ( B ) ≤ 2 ε . W e c laim that Ent ♯ ε ′ ( X ) ≤ Ent ♯ ε ( Y ). This inequality will follow as so on as we chec k that Ent δ ε ′ ( X ) < E nt ♯ ε ( Y ) for every δ < ∞ . Since the m ulti-map Φ is bo rnolog o us, there is a real n umber δ ′ such that diam (Φ( A )) ≤ δ ′ for any bo unded subset A ⊂ X with diam ( A ) ≤ 2 δ . W e claim that (9) Ent δ ε ′ ( X ) ≤ Ent δ ′ ε ( Y ) < Ent ♯ ε ( Y ) = Ent ♯ ( Y ) . THE CO ARSE CLASSIFICA TION OF HOM OGENEOUS UL TRA-METRIC SP A CES 15 The strict inequa lity follows fr om the definitio n of E nt ♯ ε ( Y ). T o prove the other inequality , take any x 0 ∈ X and observe that diam  Φ( B δ ( x 0 ))  ≤ δ ′ and thus Φ( B δ ( x 0 )) ⊂ B δ ′ ( y 0 ) for some y 0 ∈ Y . It follows that the ball B δ ′ ( y 0 ) has a n ε -net N ⊂ B δ ′ ( y 0 ) of size | N | ≤ En t δ ′ ε ( Y ). Consider the subset N 1 = { y ∈ N : dist  y , Φ( B δ ( x 0 ))  < ε } and fo r every y ∈ N 1 choose a p oint y ′ ∈ Φ ( B δ ( x 0 )) w ith dist ( y ′ , y ) < ε . Then the set N 2 = { y ′ : y ∈ N 1 } is a 2 ε -net for Φ( B δ ( x 0 )) of size | N 2 | ≤ | N 1 | ≤ | N | . F or every y ∈ N 2 pick a p o int ξ ( y ) ∈ Φ − 1 ( y ) ∩ B δ ( x 0 ). W e c la im that the set M = { ξ ( y ) : y ∈ N 2 } is a n ε ′ -net for B δ ( x 0 ). Indeed, for every a ∈ B δ ( x 0 ) a nd every b ∈ Φ( a ) we can find a p o int y ∈ N 2 with dist ( b, y ) < 2 ε . Observe that { a, ξ ( y ) } ⊂ Φ − 1 ( { b, y } ). Since diam ( { y , b } ) < 2 ε , the choice o f ε ′ guarantees that diam { a, ξ ( y ) } ≤ dia m Φ − 1 ( { b, y } ) ≤ ε ′ witnessing that M is an ε ′ -net for B δ ( x 0 ). Therefore, Ent ε ′ ( B δ ( x 0 )) ≤ | M | ≤ | N 2 | ≤ | N | ≤ Ent δ ′ ε ( Y ) and (9) holds. Now we see that Ent ♯ ( X ) ≤ Ent ♯ ε ′ ( X ) ≤ Ent ♯ ε ( Y ) = Ent ♯ ( Y ) . 2. Assume that t wo metric spaces X, Y ar e asy morphic and let Φ : X ⇒ Y b e an asymorphism. It follows fro m the preceding case that Ent ♯ ( X ) = Ent ♯ ( Y ). Now we shall prov e that ent ♯ ( X ) ≤ ent ♯ ( Y ). Find ε > 0 with ent ♯ ε ( Y ) = ent ♯ ( Y ). The borno logity of Φ − 1 yields us a real num b er ε ′ such that diam (Φ − 1 ( B )) < ε ′ for any subset B ⊂ Y of diameter diam ( B ) ≤ 2 ε . W e claim that ent ♯ ( X ) ≤ ent ♯ ε ′ ( X ) ≤ ent ♯ ε ( Y ) = ent ♯ ( Y ) . Assuming conv ersely that ent ♯ ε ′ ( X ) > ent ♯ ε ( Y ), we c o uld find δ < ∞ such that (en t δ ε ′ ( X )) + > en t ♯ ε ( Y ), whic h is equiv alent to ent δ ε ′ ( X ) ≥ ent ♯ ( Y ). The b orno logity of Φ yields a real num ber δ ′ such that diam Φ( A ) ≤ δ ′ for any subset A ⊂ X with diam ( A ) ≤ 2 δ . The definition of ent ♯ ε ( Y ) implies that min y ∈ Y Ent ε ( B δ ′ ( y )) = ent δ ′ ε ( Y ) < ent ♯ ε ( Y ) and th us there is a po int y 0 ∈ Y with E nt ε ( B δ ′ ( y 0 )) < e nt ♯ ε ( Y ). This means that the ball B δ ′ ( y 0 ) contains an ε -net N of size | N | < ent ♯ ε ( Y ). Now tak e an y p oint x 0 ∈ Φ − 1 ( y 0 ) and consider the clos e d δ - ball B δ ( x 0 ) ⊂ X . It follows from the choice o f δ ′ that diam Φ( B δ ( x 0 )) ≤ δ ′ . Since y 0 ∈ Φ( x 0 ) ⊂ Φ( B δ ( x 0 )), w e conclude that Φ( B δ ( x 0 )) ⊂ B δ ′ ( y 0 ). Rep eating the argumen t from the preceding item, we can transform the ε -net N into an ε ′ -net M ⊂ B δ ( x 0 ) o f cardinality | M | ≤ | N | . Then ent δ ε ′ ( X ) ≤ Ent ε ′ ( B δ ( x 0 )) ≤ | M | ≤ | N | < ent ♯ ε ( Y ) which is a desired contradiction that pro ves the inequality e n t ♯ ( X ) ≤ ent ♯ ( Y ). The reverse ineq uality ent ♯ ( X ) ≥ ent ♯ ( Y ) can be pr ov ed by analogy . 3. Assume that X , Y a re tw o ultra-metric spac es with Ent ♯ ( X ) ≤ ent ♯ ( Y ). Find a rea l num ber R such that Ent ♯ r ( X ) = Ent ♯ ( X ) and ent ♯ r ( Y ) = ent ♯ ( Y ) for all r ≥ R . Using the definition of E nt ♯ r ( X ) we can find t wo unbounded increa sing sequences of real num bers ~ r = ( r n ) n ∈ N and ~ ρ = ( ρ n ) n ∈ N such that r 1 = R = ρ 1 and Ent r n r n +1 ( X ) ≤ ent ρ n ρ n +1 ( Y ) for all n ∈ N . It follows from Prop os ition 7 that X is asymorphic to the ba se [ T X ( ~ r )] o f the canonic level ~ r -subtow er T X ( ~ r ) while Y is asymo rphic to the bas e [ T Y ( ~ ρ )] o f the 16 T ARAS BANAKH, IHOR ZARICHNYY level ~ ρ -subtow er T Y ( ~ ρ ) of Y . Let Φ X : X → [ T X ( ~ r )] and Φ Y : Y → [ T Y ( ~ ρ )] b e the corres p o nding asymo rphisms. Observe that Deg j i ( T X ( ~ r )) = Ent r j r i ( X ) and deg j i ( T X ( ~ r )) = ent r j r i ( X ) for all i < j . This implies that Deg n ( T X ( ~ r )) ≤ deg n ( T Y ( ~ ρ )) for a ll n ∈ N . Applying P rop osi- tion 6 we can find a tow er em bedding ϕ : T X ( ~ r ) → T Y ( ~ ρ ) which induces an isomet- ric embedding ψ = ϕ | [ T X ( ~ r )] : [ T X ( ~ r )] → [ T Y ( ~ ρ )]. Now w e see that the multi-map Ψ = Φ − 1 Y ◦ ψ ◦ Φ X : X ⇒ Y is an a symorphic embedding. Consider ed as a multi- map in to Ψ( X ) ⊂ Y , Ψ : X → Ψ( X ) is an asymor phis m of X onto the subs pa ce Ψ( X ) of Y . By Pr op osition 2, X is c oarsely equiv alent to Ψ( X ). 3. Let X b e a metric spa ce. W e need to c hec k that if En t( X ) is a limit car dinal, then it has countable cofinality . Find a real num ber ε > 0 with Ent( X ) = Ent ε ( X ) and notice that En t ε ( X ) = sup n ∈ N (Ent n ε ( X )) + . Now ass ume that κ is a cardinal κ such that either κ = 2 o r κ is an infinite successor cardinal or else κ is a limit cardinal of c ountable cofinality . W e need to find a homo geneous ultra-metric space X with Ent ♯ ( X ) = κ . F o r this we conside r 3 cases. (a) If κ ≤ ℵ 0 , then we have the necess ary examples b ecause Ent ♯ ( { 0 } ) = 2, and Ent ♯ (2 <ω ) = ℵ 0 . (b) If κ = λ + is an infinite successor cardinal, then w e can consider the ultra- metric space λ <ω and observe Ent ♯ ( λ <ω ) = λ + = κ . (c) Finally assume tha t κ is an unco untable limit cardinal of coun table co finality and choos e an increasing sequence of infinite cardinals ( κ ) n ∈ N with sup n ∈ N κ n = κ . Let X = Q ( κ ) b e a linear s pace ov er the field Q having the set of ordinals κ = { α : α < κ } for a Hamel bas is . F or every n ∈ N let L n = Q ( κ n ) b e the linear subspace algebraically g enerated b y the subset κ n ⊂ κ . On the spa ce Q ( κ ) consider the ultra-metric d ( x, y ) = 2 · max { n ∈ N : x − y / ∈ L n } where x, y ∈ X are tw o dis tinct p oints of X . Observe that for every n < m we get Ent n ( L m ) = | L m /L n | = κ m and hence Ent ♯ n ( X ) = sup m ∈ N κ + m = κ and Ent ♯ ( X ) = min n ∈ N Ent ♯ n ( X ) = κ . 7. Proof of Theorem 3. W e need to pro ve that t w o ho mogeneous ultra-metr ic spa c es X and Y are asy- morphic if and only if Ent ♯ ( X ) = Ent ♯ ( Y ). The “only if ” par t follows from Prop o- sition 1(2). T o prove the “if ” part, as sume that Ent ♯ ( X ) = Ent ♯ ( Y ) = κ . 1. If κ ≤ 1 , then the metric spaces X , Y are bounded and hence asymo rphic. 2. If κ = ℵ 0 , then the spaces X, Y , b eing homogeneous, are as ymorphic to the anti-Can tor set 2 <ω according to Theorem 2. 3. Assume that κ = µ + is an infinite successo r cardinal. Then w e can c ho ose a n un b ounded increas ing sequence ~ r = ( r n ) n ∈ N of real n um b ers such that Ent r n +1 r n ( X ) = µ = E nt r n +1 r n ( Y ) for all n ∈ N . By Pro p osition 7, X is asymor phic to the bas e [ T X ( ~ r )] o f the (homogeneous) canonic ~ r -tower T X ( ~ r ) of X . The same is true for the s pace Y : it is asymor phic to the base [ T Y ( ~ r )] o f its canonic ~ r -tower T Y ( ~ r ). By Coro lla ry 2, the homog e neous tow ers T X ( ~ r ) and T Y ( ~ r ) THE CO ARSE CLASSIFICA TION OF HOM OGENEOUS UL TRA-METRIC SP A CES 17 are isomo rphic, which implies that their bases [ T X ( ~ r )] a nd [ T Y ( ~ r )] a re isometric. Combining the as y morphisms X ∼ [ T X ( ~ r )] ∼ [ T Y ( ~ r )] ∼ Y we conclude that the spaces X , Y are asymorphic. 4. Finally assume that κ = Ent ♯ ( X ) = Ent ♯ ( Y ) is an uncountable limit ca rdinal. W e can cho ose an unbounded increasing seq uence ~ r = ( r n ) n ∈ N of real num ber s such that the sequences κ n = deg n ( T X ( ~ r )) a nd µ n = deg n ( T Y ( ~ r )), n ∈ N , consis ts of infinite cardinals, are increasing and hav e sup n ∈ N κ n = κ = sup n ∈ N µ n . In t he item 3(c) of the pro of of Theorem 3 w e defined the space Q ( κ ) endow ed with the ultrametric d 1 ( x, y ) = 2 · max { n ∈ N : x − y / ∈ Q ( κ n ) } where x, y ∈ Q ( κ ) a re distinct p oints of Q ( κ ). This space is isometric to the base of the ho mogeneous tow er T 1 = { x + Q ( κ n ) : x ∈ Q ( κ n ) , n ∈ N } with deg n ( T 1 ) = | Q ( κ n +1 ) / Q ( κ n ) | = κ n for all k ∈ N (here w e assume that κ 0 = 0). By Corollar y 2, the ho mo geneous tow ers T X ( ~ r ) and T 1 are isomor phic and consequently , their bases [ T X ( ~ r )] and Q ( κ ) = [ T 1 ] are isometric. T aking in to accoun t t hat X is asymorphic to [ T X ( ~ r )], we see that the spaces X and ( Q ( κ ) , d 1 ) are asymor phic. By the sa me reas on, Y is asymorphic to the space Q ( κ ) endow ed with the ultra- metric d 2 ( x, y ) = 2 · max { n ∈ N : x − y / ∈ Q ( µ n ) } where x, y ∈ Q ( κ ) are distinct p oints of Q ( κ ). Since the sequences ( κ n ) and ( µ n ) are s tr ictly inc r easing and hav e the same supremum, the ident ity map ( Q ( κ ) , d 1 ) → ( Q ( κ ) , d 2 ) is a bijective asymor phism. Combining the (bijective) asymorphisms X ∼ [ T X ( ~ r )] ∼ ( Q ( κ ) , d 1 ) ∼ ( Q ( κ ) , d 2 ) ∼ [ T Y ( ~ r )] ∼ Y we conclude that X and Y are asymorphic. 8. Some Open Pr oblems In this pa p er we characterized homogeneo us ultr a-metric spaces asymor phic to the anti-Cant or set: those a re exactly homoge no us ultra-metric spaces with Ent ♯ ( X ) = ℵ 0 . Howev er for arbitrary (not necessar ily homog eneous) metric spaces a similar characterization pro blem seems to be muc h more difficult. Problem 1 . Find ne c essary and sufficient c onditions on an ult r a-m et ric sp ac e X guar ant e eing that X is asy morphic to the ant i-Cantor set 2 <ω . In p articular, is X asymorphic t o 2 <ω if ent ♯ ( X ) = Ent ♯ ( X ) = ℵ 0 ? W e can p o s e a simpler question as king if the condition in Theorem 2 in volving infinite pro ducts can be r eplaced b y a weak er condition. Problem 2. Is a pr op er ultr a-metric sp ac e X asymorphic to the anti-Cantor set if ther e is a r e al c onstant C and a n incr e asing numb er se quenc e ( r i ) su ch t hat Ent r j r i ( X ) ≤ C · ent r j r i ( x ) for al l i < j ? This problem is equiv alent to the follo wing one. 18 T ARAS BANAKH, IHOR ZARICHNYY Problem 3. Is the b ase [ T ] of a pr op er tower T asymorphic to the anti-Cantor set if sup i

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