Contraction of a Generalized Metric Structure

Con tration of a Generalized Metri Struture Guy W allet Septem b er 19, 2021 Lab oratoire de Mathématiques, Image et Appliations P le Sienes et T e hnologie de l'Univ ersité de La Ro  helle A v en ue Mi hel Crép eau 17042 La Ro  helle edex 1 guy .w alletuniv-lr.fr Abstrat : In some sien ti elds, a saling is able to mo dify the top ology of an observ ed ob jet. Our goal in the presen t w ork is to in tro due a new formalism adapted to the mathematial represen tation of this kind of phenomenon. T o this end, w e in tro due a new metri struture - the galati spaes - whi h dep ends on an ordered eld extension of R . Moreo v er, some natural transformations of the ategory of galati spaes, the on trations, are of partiular in terest: they generalize usual homotheties, they ha v e a ratio whi h ma y b e an innitesimal, they are able to mo dify the top ology and they satisfy a nie omp osition rule. With the help of nonstandard extensions w e an asso iate to an y metri spae an innite family of galati spaes; lastly , w e study some limit prop erties of this family . Key w ords : Saling, on tration, generalized distane, galati spae. AMS lassiation : 03H05, 12J15, 26E35, 54E35. 1 In tro dution: saling and top ology It is w ell kno wn that the notion of sale is fundamen tal in empirial sienes: the prop erties of an ob jet generally dep end on a giv en sale and a  hange of sale (a saling) ma y deeply mo dify some of these prop erties. In some elds lik e Image Pro essing, Geographial Information Systems and Spatial Analysis, one of the ma jor eets of a saling is a p ossible mo diation of the top ology . An elemen tary example: a it y A whi h is inside a geographial area B at some sale ma y b e lo ated on the b oundary of B when onsidered at a smaller sale. A general question is to b e able to tak e in to aoun t these top ologial deformation phenomena [10 , 11 , 19 ℄. F or instane, it is a real problem to iden tify a giv en ob jet represen ted at dieren t sales. Let us notie that the salings for whi h w e hop e a determinist la w are the  ontr ations , i.e. the salings whi h derease the sale (and the size of the ob jets). The mathematial transformation naturally related to a saling is the notion of homothet y (for instane in an ane spae). But an homothet y is an homeo- morphism; th us, it lea v es in v arian t the top ology . On the basis of this observ a- tion, man y exp erts in Geographial Information Systems and Spatial Analysis 1 Con tration of a Generalized Metri Struture 2 onluded that a saling is a natural transformation whi h annot b e exatly represen ted b y a mathematial transformation. Although this opinion is the expression of a real diult y , it underestimates the mo deling apait y of math- ematis. In order to progress in our analysis, w e m ust in tro due a general mathematial framew ork adapted to this kind of problem. F or that purp ose, w e onsider the lass E of all metri spaes. Giv en a real n um b er λ > 0 , the homothet y of ratio λ is the quit trivial transformation op erating on E whi h maps ea h metri spae E := ( E , d ) to the new one λE := ( E , λd ) . W e notie that w e ha v e the nie omp osition prop ert y λ ( µE ) = ( λµ ) E and more generally , that the homotheties result from the left ation ( λ, E ) 7→ λE of the m ultipliativ e group R ∗ + on E . Nev ertheless, sine the distane d and λd dene the same top ology on the set E , w e nd again the in v ariane prop ert y of top ology b y an homothet y . Th us, it is still true that an homothet y is not a go o d represen tativ e of a saling. On the other side, w e an also agree that an empirial saling is m u h more that a simple  hange of size. In realit y , a  onr ete s aling se ems to b e the union of two distint but dep endent pr o  esses: (1) an homothety whih hanges, p ossibly very str ongly, the size of any obje t, (2) a simpli ation whih al lows to ne gle t to o smal l details . In order to build a on v enien t mathematial onept of sal- ing, w e ha v e to translate sim ultaneously these t w o pro esses in an appropriate mathematial notion. A tually , a ma jor w ork w as already done on this topi. It is ab out the limit of a sequene ( λ n E ) in E for the Gromo v-Hausdor dis- tane where E is a giv en metri spae and ( λ n ) is a sequene in R ∗ + su h that λ n → 0 . In tro dued b y Gromo v in his study on group of p olynomial gro wth in 1981 [5, 6 , 7 ℄, this onept ollets the t w o main asp ets of a onrete saling: the sequene ( λ n ) orresp onds to the strong homothet y (atually a strong on- tration) and the limit orresp onds to the simpliation pro ess. Nev ertheless, it is not easy to handle with this kind of limit b eause there are few general results of on v ergene for a sequene of the t yp e ( λ n E ) . In 1984 [18 ℄, V an den Dries and Wilkie dened a non standard alternativ e: the asymptoti one of a metri spae E relativ e to a en ter x 0 in a nonstandard extension ∗ E of E and for a similitude ratio λ whi h is no w an innitesimal h yp erreal n um b er. The main adv an tage of this onstrution is that its existene is ertain, whatev er b e E , λ and x 0 ; its strongest disadv an tage is its transenden t  harater due to its dep endane to a non trivial ultralter. Sine their in tro dution, these onepts ha v e b een the sub jet of man y deep and in teresting w orks at the b order of group theory , top ology and logi (for instane [2, 3, 12 , 4 ℄). Nev ertheless, w e notie that, in the framew ork of this t w o approa hes, it is diult to form ulate and giv e a simple meaning to a omp osition rule of t w o saling. Our study is not stritly in the eld of the preeding w orks on asymptoti ones. Indeed, our main goal is to bring a p ositiv e answ er to the follo wing question. Is it p ossible to gener alize the notion of homothety so as to get a lass S of tr ansformations op er ating on a lass G of sp a es with the pr op erties set out b elow? 1. Eah sp a e G ∈ G is pr ovide d with a kind of metri strutur e whih is a gener alization of the strutur e of a metri sp a e. 2. Eah tr ansformation s ∈ S mo dify (p ossibly very str ongly) the size of any obje t and the str ength of this mo di ation is me asur e d by a r ation λ s . Con tration of a Generalized Metri Struture 3 3. The underlying top olo gi al strutur e is p ossibly alter e d by suh op er ations. 4. F or e ah s, s ′ ∈ S , ther e is the ni e  omp osition rule λ s ◦ s ′ = λ s .λ s ′ . The base of the presen t w ork is the observ ation that the onstrution of a non- standard asymptoti one atually pro dues an in termediary spae whi h arries a more general metri struture than the struture of metri spae. W e all this new struture a galati spae. A remark able feature is that the notion of gala- ti spae is not stritly dep enden t of the nonstandard framew ork: to dene it, w e only need an ordered eld extension of R . Moreo v er, some natural trans- formations of the ategory of galati spaes, the on trations, are of partiular in terest: they generalize usual homotheties but they ha v e a ratio whi h ma y b e an innitesimal, they arry out a simpliation pro ess and they satisfy a nie omp osition rule. Th us, at the lev el of galati spaes, the on trations seem to b e go o d represen tativ es of salings. It seems probable that partially similar strutures ha v e already b een used in other on texts. F or instane, this is the ase for the basi notion of distane taking its v alues in a ordered quotien t group. It is only at the time he w as nish- ing the nal bibliograph y of this pap er that the author found in [13 ℄ the notion of ultraone whi h is almost an example of the general struture of galati spae. 2 Innitesimals in an extension eld of R 2.1 In all this study , w e onsider a pr op er or der e d eld extension K of the eld R of real n um b ers. Th us, K is an ordered eld, R ( K and the restrition to R of the ordered eld struture of K is the usual ordered eld struture or R . W e ma y think to some algebrai examples su h as the eld R ( X ) of rational frations R ( X ) :=  P ( X ) Q ( X ) ; ( P ( X ) , Q ( X )) ∈ R [ X ] 2 and Q ( X ) 6 = 0  or the eld R (( X )) of Lauren t p o w er series R (( X )) := ( + ∞ X i = m a i X i ; m ∈ Z and ∀ i ≥ m a i ∈ R ) or the eld of Puiseux series with real o eien ts R [[ X Q ]] := [ n ≥ 1 R (( X 1 /n )) b oth pro vided with the order relation for whi h 0 < X < 1 /n for all n ∈ N ∗ . Another in tersting example arises from nonstandard analysis: a eld ∗ R or h yp erreal n um b ers [14 , 16 ℄. The simplest w a y to onstrut a eld of h yp erreal n um b ers is to tak e an ultra-p o w er of R . T o this end, w e  ho ose a non prinipal ultralter U on the set N of natural n um b ers (0 inluded). That means that U is a family of subsets of N su h that Con tration of a Generalized Metri Struture 4 1. U 6 = ∅ and ∅ 6∈ U , 2. ∀ ( U, V ) ∈ P ( N ) ( U ∈ U and U ⊂ V = ⇒ V ∈ U ) , 3. ∀ ( U, V ) ∈ U U ∩ V ∈ U , 4. ∀ F ∈ P ( N ) ( F nite = ⇒ F 6∈ U ) . Then, w e onsider the set R N of sequenes of real n um b ers and the equiv alene relation ∼ U on R N su h that ( x n ) ∼ U ( y n ) if and only if { n ∈ N ; x n = y n } ∈ U . The set ∗ R of h yp erreal n um b ers is the quotien t set R N / ∼ U . It is easy to  he k diretly (whithout using an y sp eial logial to ol) that ∗ R is an ordered eld for the natural op erations and the order relation [( x n )] U ≤ [( y n )] U ⇐ ⇒ { n ∈ N ; x n ≤ y n } ∈ U where [( x n )] U and [( y n )] U are the equiv alene lasses of the sequenes ( x n ) and ( y n ) . The map ∗ : R → ∗ R su h that the image of an y x ∈ R is the equiv alene lass of the sequene of onstan t v alue x is learly a eld morphism whi h preserv es order relations. Th us, ∗ R is an ordered eld extension of R . Moreo v er, the lass of the sequene (0 , 1 , 2 , . . . , n, . . . ) do es not b elong to the image of R in ∗ R . Consequen tly , this extension is prop er. No w, w e onsider the general ase of a prop er ordered extension K or R . Let s an elemen t of K ; then s is innitely smal l and w e write s ≃ 0 if | s | < 1 n for ea h n ∈ N ∗ ; s is innitely lar ge if n < | s | for ea h n ∈ N (w e write s ≃ + ∞ if s > 0 and s ≃ − ∞ if s < 0 ); s is limite d if it is not innitely large. W e see that the in v erse of an innitely large elemen t is an innitely small elemen t. Prop osition 1. In K , ther e ar e innitely lar ge elements and non nul l innitely smal l elements. Mor e over, for e ah s ∈ K whih is not innitely lar ge, ther e exists one and only one o s ∈ R suh that s ≃ o s . Pr o of. Let s ∈ K \ R . If s is innitely large, the rst p oin t is pro v en. If not, w e are under the assumptions of the seond p oin t. Then, w e onsider the ut ( I , S ) of R dened b y I = { x ∈ R ; x < s } and S = { x ∈ R ; x > s } . This ut denes one and only one real n um b er o s su h that o s = sup I and o s = inf S . Then it is neessary that s ≃ o s and w e see that s − o s is innitely small and dieren t from 0. 2.2 The halo of 0 is the set Hal (0) whose elemen ts are the innitely small elemen ts of K . It is an additiv e subgroup of K to whi h is asso iated the pro ximit y relation ≃ dened b y ∀ ( s, t ) ∈ K 2 ( s ≃ t ⇐ ⇒ s − t ∈ Hal (0)) The halo of an y x ∈ K is the equiv alene lass Hal ( x ) of x for ≃ , that is to sa y Hal ( x ) = x + Hal (0) . The quotien t group Hal ( K ) := K / Hal (0) is the set of all Hal ( x ) for x ∈ K and the map Hal : x 7→ Hal ( x ) is the anonial pro jetion of K on Hal ( K ) . The galaxy of 0 is the set Gal (0) whose elemen ts are the limited elemen ts of K . It is lear that Hal (0) ⊂ Gal (0) and that Gal (0) is also an additiv e subgroup of K . F rom the preeding prop osition, w e dedue that Gal (0) / Hal (0) is isomorphi Con tration of a Generalized Metri Struture 5 to R and Gal (0) is equal to the disjoin t union [ x ∈ R Hal ( x ) . Moreo v er, there exists a prinip al value map p v : K → R ∪ { + ∞ , −∞} su h that ∀ s ∈ K p v ( s ) =    o s if s ∈ Gal (0) (where o s ∈ R and o s ≃ s ) + ∞ if s ≃ + ∞ −∞ if s ≃ −∞ so that s ≃ p v ( s ) for ev ery s ∈ K . If K is a eld of nonstandard h yp erreal n um b ers ∗ R , the map p v is usually alled the standar d p art map and denoted st. Due to the fat that the sum of t w o innitesimals is an innitesimal, w e see that p v ( s + t ) = p v ( s ) + p v ( t ) for ev ery t, s ∈ K su h that s, t 6≃ ±∞ . Let us onsider the quotien t group Gal ( K ) := K / Gal (0) and the anonial pro- jetion Gal : K → Gal ( K ) . F or ea h t ∈ K , the equiv alene lass Gal ( t ) = t + Gal (0) is alled the galaxie of t . There is a natural total order relation ≤ on Gal ( K ) dened b y ∀ ( s, t ) ∈ K 2 [ Gal ( s ) ≤ Gal ( t ) ⇐ ⇒ ( s ≤ t or s − t ∈ Gal (0))] Moreo v er, this relation is ompatible with the additiv e struture of Gal ( K ) ∀ ( s, t ) ∈ K 2 [(0 ≤ Gal ( s ) and 0 ≤ Gal ( t )) = ⇒ 0 ≤ Gal ( s ) + Gal ( t )] These prop erties mean that Gal ( K ) is an ordered additiv e group. The terms of halo and galaxy are already used in some dev elopmen t of nonstan- dard analysis; in this ase, they denotes t w o imp ortan t lasses of external sets [1, 17 ℄. Our study is not diretly in onnetion with these sp ei prop erties. In addition, it is probable there are similar onepts in other on texts using non-ar himedean extensions. 3 Galati spae Giv en an y ordered additiv e group G , w e denote b y G + and G ∗ + the follo wing sets G + = { x ∈ G ; x ≥ 0 } and G ∗ + = { x ∈ G ; x > 0 } . The sym b ol + ∞ is supp osed su h that, for an y x ∈ G , w e ha v e x < + ∞ and x + (+ ∞ ) = + ∞ + x = + ∞ . Giv en a map d dened on a pro dut X × X and taking its v alues in G + or G + ∪ { + ∞} , w e sa y that d satises the gener al metri rule if , for ev ery x, y , z ∈ X 1. d ( x, x ) = 0 2. d ( x, y ) = d ( y , x ) > 0 for x 6 = y 3. d ( x, z ) ≤ d ( x, y ) + d ( y , z ) Hene, a metri spae is a struture ( X, d ) su h that X is a set and d is a a map d : X × X → R + whi h satises the general metri rule (i.e. d is a distane on X ). Denition 1. Given a set F , a map δ : F × F → R + ∪ { + ∞} whih satises the gener al metri rule is  al le d a generalized distane on F . Con tration of a Generalized Metri Struture 6 If δ is a generalized distane on F , for ea h ( x, r ) ∈ F × ( R + ∪ { + ∞} ) , w e an dene the op en ball of en ter x and radius r B δ ( x, r ) = { y ∈ F ; δ ( x, y ) < r } The op en balls of radius + ∞ are also the equiv alene lass for the relation δ ( x, y ) < + ∞ . Ea h large op en ball is learly a metri spae for δ and is alled a metri  omp onent of F for δ . Let M F b e the set of all metri omp onen ts of F for δ and, for ea h x ∈ F , let C F ( x ) the elemen t E ∈ M F su h that x ∈ E . If δ is a generalized distane on a set F , then the family of op en balls is a basis of a top ology on F . F urthermore, F is the disjoin t union of its metri omp onen ts and ea h metri omp onen t is an op en set of F . Reipro ally , if ( E i , δ i ) i ∈ I is a family of disjoin ts metri spaes, then ( E i ) i ∈ I is the family of metri omp onen ts of F = S i ∈ I E i for the generalized distane δ dened b y ∀ ( x, y ) ∈ F 2 δ ( x, y ) =  δ i ( x, y ) if ∃ i ∈ I su h that x, y ∈ E i , + ∞ else. Consequen tly , a set pro vided with a generalized distane is just the disjoin t union of a family of metri spaes. W e w an t to impro v e this onept b y the onsideration of a kind of metri on the set of metri omp onen ts. Let K b e a xed ordered extension eld of R . Therefore, w e ha v e the ordered group Gal ( K ) whose elemen ts are the galaxies of K . Denition 2. A galati distane on a set E is a map ∆ : E × E → Gal ( K ) whih satises the gener al metri rule. If ∆ is a galati distane on a set E , there is a w ell dened top ology on E so that the family of op en balls is a basis of this top ology . Denition 3. A galati spae is a strutur e ( F, δ, ∆) in whih F is a set, δ is a gener alize d distan e on F and ∆ is a galati distan e on the set M F of metri  omp onents of F for δ . With the aim of simplifying the notations, w e an also sa y that F is a galati spae without men tioning δ and ∆ . In some w a y , a galati spae is a set with t w o lev els of resolution: a ne resolution giv en b y the generalized distane whi h relates the top ologial relations b et w een p oin ts inside ea h metri omp onen t, a oarse resolution giv en b y the galati distane whi h relates the top ologial relations b et w een the metri omp onen ts. In spite of the  hosen terminology , the reader m ust a v oid to think that the struture of galati spae ma y ha v e an y appliation in the siene of univ erse. Denition 4. A n isometry b etwe en two galati sp a es ( F, δ, ∆) and ( F ′ , δ ′ , ∆ ′ ) is a bije tive map φ : F → F ′ suh that 1. ∀ ( x, y ) ∈ F 2 δ ′ ( φ ( x ) , φ ( y )) = δ ( x, y ) 2. ∀ ( G, H ) ∈ M 2 F ∆ ′ ( φ ( G ) , φ ( H )) = ∆( G, H ) Con tration of a Generalized Metri Struture 7 (It is lear that, if φ is bijetiv e and satises the p oin t (1.), then, for ea h metri omp onen t G of F for δ , the set φ ( G ) is a metri omp onen t of F ′ for δ ′ .) Example 1 A metri spae ( E , d ) is a partiular ase of galati spae, that is to sa y the galati spae ( E , d, D ) where D is the trivial galati distane on the set { E } . This trivial example sho ws that the notion of galati spae is really a generalization of that of metri spae. Example 2 Let ( E 1 , d 1 ) and ( E 2 , d 2 ) t w o metri spaes su h that E 1 ∩ E 2 = ∅ . W e  ho ose a galaxy ∆ 1 , 2 ∈ Gal ( K ) whi h is not trivial ( ∆ 1 , 2 6 = 0 ∈ Gal ( K ) ). Then, it is easy to v erify that there is a galati spae ( E , d, D ) su h that • E = E 1 ∪ E 2 ; • d ( x, y ) =    d 1 ( x, y ) if ( x, y ) ∈ E 1 2 d 2 ( x, y ) if ( x, y ) ∈ E 2 2 + ∞ else ; • the metri omp onen ts of E for d are E 1 and E 2 and D ( E 1 , E 2 ) = ∆ 1 , 2 . Example 3 Let ( G, d ) b e a K -metri sp a e : that means that G is a set and d is a map from G × G to K + whi h satises the general metri rule. The simplest example of a K -metri spae is G = K and d ( x, y ) = | x − y | for ea h ( x, y ) ∈ K 2 . Another example is G = ∗ E and d = ∗ d where ( E , d ) is a metri spae, ( ∗ E , ∗ d ) is a nonstandard extension of ( E , d ) and K = ∗ R . Then, w e onsider the equiv alene relation ≈ on G dened b y ∀ ( x, y ) ∈ G 2 ( x ≈ y ⇔ d ( x, y ) ≃ 0) and the quotien t set F = G/ ≈ . F or ev ery x ∈ G , w e denote [ x ] the equiv alene lass of x for ≈ . On F w e ha v e a generalized distane δ dened b y ∀ ( x, y ) ∈ G 2 δ ([ x ] , [ y ]) = p v ( d ( x, y )) Finally , w e dene a galati distane ∆ on the set M F of metri omp onen ts of F for δ su h that ∀ ( x, y ) ∈ G 2 ∆( C F ( x ) , C F ( y )) = Gal ( d ( x, y )) where C F ( t ) denotes the metri omp onen t of t ∈ F . Then ( F, δ, ∆) is a galati spae. W e sa y that ( F, δ, ∆) is the galati pr oje tion of the K -metri spae ( G, d ) . The follo wing result sho ws that the preeding example is univ ersal. Theorem 1. Every galati sp a e is the galati pr oje tion of a K -metri sp a e. Pr o of. W e onsider a galati spae ( F, δ, ∆) and let M F b e the set of metri omp onen ts of this spae. F or ea h ( E i , E j ) ∈ M 2 F su h that E i 6 = E j , w e  ho ose an elemen t d ij in the galaxy ∆( E i , E j ) in su h a w a y that d ij = d j i . Th us, w e ha v e d ij ≃ + ∞ in K and Gal ( d ij ) = ∆( E i , E j ) . In the same w a y , Con tration of a Generalized Metri Struture 8 w e  ho ose an elemen t x i in ea h metri omp onen t E i . Then, w e dene a map d : F 2 → K + su h that, for ea h ( x, y ) ∈ F 2 d ( x, y ) =  δ ( x, y ) if δ ( x, y ) < + ∞ δ ( x, x i ) + d ij + δ ( x j , y ) if ∃ ( E i , E j ) ∈ M 2 F E i 6 = E j ( x, y ) ∈ E i × E j . W e see at one that d is symmetrial. Sine d ( x, y ) ≃ + ∞ for ev ery ( x, y ) ∈ E i × E j su h that E i 6 = E j , w e ha v e ∀ ( x, y ) ∈ F 2 ( d ( x, y ) = 0 ⇐ ⇒ x = y ) It remains to pro v e the triangular inequalit y d ( x, z ) ≤ d ( x, y ) + d ( y , z ) for ea h ( x, y , z ) dans F 3 . If x, y , z b elong to the same metri omp onen t, there is no problem. Th us, w e ha v e to onsider four ases. Case 1 : x ∈ E i and y , z ∈ E j with E i 6 = E j in M F . Then, d ( x, z ) = δ ( x, x i ) + d ij + δ ( x j , z ) , d ( x, y ) = δ ( x, x i ) + d ij + δ ( x j , y ) and d ( y , z ) = δ ( y , z ) . Hene, the result ome from δ ( x j , z ) ≤ δ ( x j ) , y ) + δ ( y , z ) . Case 2 : x, y ∈ E i and z ∈ E j with E i 6 = E j in M F . Then, d ( x, z ) = δ ( x, x i ) + d ij + δ ( x j , z ) , d ( x, y ) = δ ( x, y ) and d ( y , z ) = δ ( y , x i ) + d ij + δ ( x j , z ) . Hene, the result omes from δ ( x, x i ) ≤ δ ( x, y ) + δ ( y , x i ) . Case 3 : x, z ∈ E i and y ∈ E j with E i 6 = E j in M F . Then, d ( x, z ) = δ ( x , z ) , d ( x, y ) = δ ( x , x i ) + d ij + δ ( x j , y ) and d ( y , z ) = δ ( y , x j ) + d j i + δ ( x i , z ) . Sine δ ( x, z ) ∈ R + , d ( x, y ) ≃ + ∞ and d ( y , z ) ≃ + ∞ , w e get the result. Case 4 : x ∈ E i , y ∈ E j and z ∈ E k with E i 6 = E j , E i 6 = E k and E j 6 = E k in M F . Then, d ( x, z ) = δ ( x, x i ) + d ik + δ ( x k , z ) , d ( x, y ) = δ ( x, x i ) + d ij + δ ( x j , y ) and d ( y , z ) = δ ( y , x j ) + d j k + δ ( x k , z ) . Sine ∆( E i , E k ) ≤ ∆( E i , E j ) + ∆( E j , E k ) in Gal ( K ) = K / Gal (0) , w e ha v e r ik ≤ r ij + r j,k for ev ery r ik ∈ ∆( E i , E k ) , r ij ∈ ∆( E i , E j ) and r j k ∈ ∆( E j , E k ) . Hene, the result omes from d ( x, z ) ∈ ∆( E i , E k ) , d ( x, y ) ∈ ∆( E i , E j ) and d ( y , z ) ∈ ∆( E j , E k ) . No w, w e kno w that ( F, d ) is a K -metri spae. It is easy to  he k that the galati pro jetion of ( F, d ) is isomorphi to ( F, δ, ∆) . F rom the preeding result, w e ould hastily onlude that the study of galati spaes ma y b e adv an tageously replaed b y the study of K -metri spaes. On the on trary , w e think that galati spaes are in teresting b eause they ha v e a ri h metri struture whi h is a suitable framew ork for saling. In the next setion, w e will in tro due a notion of on tration whi h naturally op erates on the lass of galati spaes. Example 4 Let us all galati  ontinuous line the galati spae D c, K whi h is the galati pro jetion of K view as a K -metri spae. Th us, the set D c, K is equal to Hal ( K ) = K / Hal (0) , the generalized distane is giv en b y ∀ ( x, y ) ∈ K δ ( Hal ( x ) , Hal ( y )) = vp ( | x − y | ) , Con tration of a Generalized Metri Struture 9 for ea h x ∈ K the metri omp onen t of Hal ( x ) is G ( x ) := { Hal ( y ) ; y ∈ Gal ( x ) } , the set of metri omp onen ts is the quotien t group of Hal ( K ) b y the subgroup Gal (0) / Hal (0) (quotien t whi h is anonially isomorphi to Gal ( K ) ) and the galati distane is giv en b y ∀ ( x, y ) ∈ K 2 ∆( G ( x ) , G ( y )) = Gal ( | x − y | ) . Sine the prinipal v alue map is bijetiv e if onsidered as a map from Gal (0) / Hal (0) to R and sine G ( x ) = Hal ( x ) + G (0) , w e see that ea h metri omp onen t of D c, K is a metri spae isometri to R and D c, K is the disjoin t union of a family ( R C ) C ∈ Gal ( K ) of opies of R . In the same w a y , w e dene a galati disr ete line D d, K to b e the disjoin t union of a family ( Z C ) C ∈ Gal ( K ) where ea h Z C is a op y of the set Z of in tegers. On D d, K , w e onsider a generalized distane δ su h that ∀ x, y ∈ D d, K δ ( x, y ) =  | x − y | if ∃ C ∈ Gal ( K ) su h that ( x, y ) ∈ C 2 + ∞ else The metri omp onen ts of D d, K for δ are the sets Z C for C ∈ Gal ( K ) and the galati distane is simply dened b y ∀ C, C ′ ∈ Gal ( K ) ∆( Z C , Z C ′ ) = | C − C ′ | . The disrete galati line is the galati spae ( D d, K , δ, ∆) . 4 Con tration of a galati spae 4.1 Before going further, w e need some onsiderations on the m ultipliation of a galaxy b y a n um b er. Let γ ∈ K ∗ + b e a limited n um b er, τ ∈ Gal ( K ) and t ∈ K su h that τ = Gal ( t ) . As usual, w e dene the set γ .τ = { γ s ; s ∈ τ } so that γ .τ = γ t + γ Gal (0) . Th us, γ .τ = Gal ( γ t ) if γ is appreiable and γ .τ ⊂ Hal ( γ t ) ⊂ Gal ( γ t ) if γ ≃ 0 . Then, w e dene γ • τ in Gal ( K ) b y γ • τ := Gal ( γ t ) ; hene, w e alw a ys ha v e γ .τ ⊂ γ • τ . Moreo v er, if τ ∈ Gal ( K ) ∗ + , the prinipal v alue map is onstan t on the set γ .τ and this onstan t v alue is named p v ( γ .τ ) . Indeed, su h a τ an b e written Gal ( t ) for some t ∈ K + su h that t ≃ + ∞ ; therefore ev ery elemen t of γ .τ is innitely large if γ 6≃ 0 and γ .τ ⊂ Hal ( γ t ) if γ ≃ 0 . 4.2 Let ( F, δ, ∆) and ( F ′ , δ ′ , ∆ ′ ) b e t w o galati spaes and let f b e a map F → F ′ . W e supp ose that f satises a Lips hitz ondition for ( δ, δ ′ ) , that is to sa y , there is a limited elemen t γ > 0 of K su h that ∀ ( x 1 , x 2 ) ∈ F 2 δ ′ ( f ( x 1 ) , f ( x 2 )) ≤ γ δ ( x 1 , x 2 ) When one of its mem b ers is + ∞ , this inequalit y m ust b e in terpreted aording to the follo wing usual rule: ∀ α ∈ K α ≤ + ∞ and α (+ ∞ ) = + ∞ , . If f satises su h a ondition, then ∀ ( x 1 , x 2 ) ∈ F 2 ( δ ( x 1 , x 2 ) < + ∞ = ⇒ δ ′ ( f ( x 1 ) , f ( x 2 )) < + ∞ ) Con tration of a Generalized Metri Struture 10 Th us, for ev ery metri omp onen t E of F for δ , the diret image f ( E ) is a subset of a metri omp onen t of F ′ for δ ′ whi h w e denote f [ E ] . Hene, w e get a map b et w een the sets of metri omp onen ts e f : M F − → M F ′ E 7− → f [ E ] Denition 5. Given two galati sp a es ( F, δ, ∆) and ( F ′ , δ ′ , ∆ ′ ) , a morphism fr om ( F, δ, ∆) to ( F ′ , δ ′ , ∆ ′ ) is a map f : F → F ′ suh that ther e is a limite d element γ > 0 in K so that the fol lowing Lipshitz  onditions ar e satise d 1. ∀ ( x 1 , x 2 ) ∈ F 2 δ ′ ( f ( x 1 ) , f ( x 2 )) ≤ γ δ ( x 1 , x 2 ) ; 2. ∀ ( E 1 , E 2 ) ∈ ( M F ) 2 ∆ ′ ( e f ( E 1 ) , e f ( E 2 )) ≤ γ • ∆( E 1 , E 2 ) ; The element γ is  al le d the Lipshitz  onstant of f . A n isomorphism fr om ( F, δ, ∆) to ( F ′ , δ ′ , ∆ ′ ) is a morphism f : ( F, δ, ∆) → ( F ′ , δ ′ , ∆ ′ ) suh that f : F → F ′ is bije tive and f − 1 is a morphism fr om ( F ′ , δ ′ , ∆ ′ ) to ( F, δ, ∆) . W e remark that if f : ( F, δ, ∆) → ( F ′ , δ ′ , ∆ ′ ) is an isomorphism, then f [ E ] = f ( E ) for ea h E ∈ M F . An isometry is a partiular ase of isomorphism b et w een t w o galati spaes. Hene, w e ha v e dened a ategory G K whose ob jets are the galati spaes pro vided with the morphisms dened just ab o v e. There is a lass of morphisms whi h is partiularly in teresting for the study of salings. Denition 6. Given a limite d element γ > 0 in K , a γ -on tration of ( F, δ, ∆) is a morphism of galati sp a es f : ( F, δ, ∆) → ( F ′ , δ ′ , ∆ ′ ) suh that f is a surje tive map fr om F to F ′ and, ∀ ( x 1 , x 2 ) ∈ F 2 ∀ ( E 1 , E 2 ) ∈ M 2 F δ ′ ( f ( x 1 ) , f ( x 2 )) =  p v ( γ δ ( x 1 , x 2 )) if δ ( x 1 , x 2 ) < + ∞ p v ( γ . ∆( C F ( x 1 ) , C F ( x 2 ))) if δ ( x 1 , x 2 ) = + ∞ ∆ ′ ( f [ E 1 ] , f [ E 2 ]) = γ • ∆( E 1 , E 2 ) The element γ is  al le d the o eien t of the  ontr ation f . W e also sa y that a galati spae ( F ′ , δ ′ , ∆ ′ ) is a γ -on tration of ( F, δ, ∆) if there exists a morphism f : ( F, δ, ∆) → ( F ′ , δ ′ , ∆ ′ ) whi h is a γ -on tration. W e notie that a 1-on tration is an isometry . It is easy to  he k that, if a surjetiv e map f : F → F ′ satises the t w o last onditions of the preeding denition, then f is neessary a morphism from ( F, δ, ∆) to ( F ′ , δ ′ , ∆ ′ ) . W e p oin t out that the o eien t γ of a on tration ma y b e greater that 1, but not to m u h. 4.3 Let us onsider the rst prop erties of on trations. Prop osition 2. L et us  onsider a limite d element γ > 0 in K and a γ -  ontr ation f γ : ( F, δ, ∆) → ( F γ , δ γ , ∆ γ ) . 1. If γ 6≃ 0 , then the map f γ : F → F γ is a bije tion and, for every x, y ∈ F δ γ ( f γ ( x ) , f γ ( y )) = pv ( γ ) δ ( x, y ) ∆ γ ( C F γ ( f γ ( x )) , C F γ ( f γ ( y ))) = γ • ∆( C F ( x ) , C F ( y )) Con tration of a Generalized Metri Struture 11 2. If γ ≃ 0 , then the map f γ is a surje tive map suh that, for every u ∈ F , we have f − 1 γ ( f γ ( u )) = { v ∈ F ; γ . ∆( C F ( u ) , C F ( v )) ⊂ Hal (0) } ; furthermor e, for every x, y ∈ F δ γ ( f γ ( x ) , f γ ( y )) = pv ( γ . ∆( C F ( x ) , C F ( y ))) ∆ γ ( C F γ ( f γ ( x )) , C F γ ( f γ ( y ))) = γ • ∆( C F ( x ) , C F ( y )) Pr o of. Straigh tforw ard. Theorem 2. L et ( F, δ, ∆) b e a galati sp a e and a limite d element γ > 0 in K . Then, ther e is a γ - ontr ation f γ : ( F , δ, ∆) → ( F γ , δ γ , ∆ γ ) of ( F, δ, ∆) . Pr o of. Let us onsider the equiv alene relation ∼ γ on F su h that, for all ( x, y ) ∈ F 2 x ∼ γ y ⇐ ⇒  γ δ ( x, y ) ≃ 0 if δ ( x, y ) < + ∞ γ . ∆( C F ( x )) , C F ( y )) ⊂ Hal (0) if δ ( x, y ) = + ∞ Let F γ b e the quotien t set F / ∼ γ and f γ : F → F γ b e the anonial pro jetion. Then, w e dene a map δ γ : F γ × F γ → R so that, for all ( x, y ) ∈ F 2 δ γ ( f γ ( x ) , f γ ( y )) =  p v ( γ δ ( x, y )) if δ ( x, y ) < + ∞ p v ( γ . ∆( C F ( x ) , C F ( y ))) if δ ( x, y ) = + ∞ whi h is learly a generalized distane on F γ . In a similar w a y , w e dene a galati distane ∆ γ on the set M F γ of metri omp onen ts of F γ for δ γ su h that ∀ ( x, y ) ∈ F 2 ∆ γ ( C F γ ( f γ ( x )) , C F γ ( f γ ( y ))) = γ • ∆( C F ( x ) , C F ( y )) Then f γ : ( F, δ, ∆) → ( F γ , δ γ , ∆ γ ) is learly a γ -on tration of ( F, δ, ∆) . This pro of sho ws that, giv en a galati spae F and γ , the onstrution of a γ -on tration of F is obtained b y a relativ ely expliit pro edure of quotien t. 4.4 Example:  ontr ation of the  ontinuous and the disr et galati lines. Giv en the on tin uous galati line D c, K = ( Hal ( K ) , δ, ∆) and an innitesimal γ su h that 0 < γ , w e w an t to onstrut a γ -on tration of D c, K . T o this end, w e onsider the additiv e subgroup γ − 1 . Hal (0) := { γ − 1 x ; x ∈ Hal (0) } of K , the quotien t group γ − 1 -Hal ( K ) := K /γ − 1 . Hal (0) whose elemen ts are the sets γ − 1 - Hal ( x ) := x + γ − 1 . Hal (0) for x ∈ K . On γ − 1 -Hal ( K ) , w e dene the gener- alized distane δ ′ b y ∀ x, y ∈ K δ ′ ( γ − 1 -Hal ( x ) , γ − 1 - Hal ( y )) = p v ( γ | x − y | ) . Ea h metri omp onen t is of the form γ − 1 -Gal ( x ) := x + γ − 1 . Gal (0) for x ∈ K and the set of metri omp onen ts is the quotien t group γ − 1 -Gal ( K ) := K /γ − 1 . Gal (0) . Then, w e onsider the galati distane ∆ ′ dened on γ − 1 -Gal ( K ) b y ∀ x, y ∈ K ∆ ′ ( γ − 1 -Gal ( x ) , γ − 1 -Gal ( y )) = Gal ( γ | x − y | ) . Hene, the map f ′ : Hal ( K ) − → γ − 1 - Hal ( K ) Hal ( x ) 7− → γ − 1 -Hal ( x ) Con tration of a Generalized Metri Struture 12 is learly a γ -on tration of D c, K . Sine f ′ is not injetiv e (for instane f ′ (0) = f ′ (1) ), this map is not an isometry . Nev ertheless, the map Hal ( K ) − → γ − 1 - Hal ( K ) Hal ( x ) 7− → γ − 1 -Hal ( γ x ) is an isometry of the galati spae D c, K to its γ -on tration. Th us, a γ -  ontr ation of a  ontinuous galati line is isometri to itself . Let us no w onsider the ase of the disrete galati line D d, K . T o this end, for ea h C ∈ Gal ( K ) w e arbitrarily  ho ose an elemen t x C ∈ C and w e dene the map f ′′ : D d, K → γ − 1 - Hal ( K ) su h that, for ea h x ∈ D d, K , w e ha v e f ′′ ( x ) = γ − 1 - Hal ( x C ) where x ∈ Z C . Then, it is lear that f ′′ is a γ -on tration from ( D d, K , δ, ∆) to ( γ − 1 -Hal ( K ) , δ ′ , ∆ ′ ) . Sine this last galati spae is isometri to D c, K , w e see that for γ ≃ 0 , a γ -s aling of the disr ete galati line is isometri to the  ontinuous galati line. 4.5 No w, w e w an t to understand the relations b et w een the dieren t on trations of a giv en galati spae. Prop osition 3. W e  onsider two limite d elements α, β ∈ K suh that 0 < β ≤ α and two morphisms of galati sp a es f α : ( F , δ, ∆) → ( F α , δ α , ∆ α ) and f β : ( F , δ, ∆) → ( F β , δ β , ∆ β ) dene d on the same sp a e suh that f α is an α - ontr ation and f β is a β - ontr ation. Then, ther e is an unique map f β ,α : F α → F β suh that f β = f β ,α ◦ f α . Mor e over, f β ,α is a β /α - ontr ation ( F α , δ α , ∆ α ) → ( F β , δ β , ∆ β ) . W e sa y that f β ,α is the tr ansition b et w een the t w o on trations ( F α , δ α , ∆ α ) and ( F β , δ β , ∆ β ) of ( F, δ, ∆) . Pr o of. F or all ( x 1 , x 2 ) ∈ F , if δ α ( f α ( x 1 ) , f α ( x 2 )) = 0 then δ β ( f β ( x 1 ) , f β ( x 2 )) = 0 ; sine f α is surjetiv e, w e dedue that there exists an unique map f β ,α : F α → F β su h that f β = f β ,α ◦ f α . It is easy to  he k that f β ,α is a β /α on tration. Corollary 1. F or e ah limite d element γ > 0 in K , two γ - ontr ations of a same galati sp a e ar e isometri. If a galati spae ( F 0 , δ 0 , ∆ 0 ) is su h that the F 0 has only one elemen t, then δ 0 and ∆ 0 are trivial and w e sa y that ( F 0 , δ 0 , ∆ 0 ) is a trivial galati spae. W e notie that, for ea h galati spae ( F, δ, ∆) , there is an unique morphism from ( F, δ, ∆) to ( F 0 , δ 0 , ∆ 0 ) . The next result sho ws that the limit when γ → 0 of the γ -on trations of a galati spae is a trivial galati spae. Prop osition 4. L et us  onsider a galati sp a e ( F, δ, ∆) and a family of gala- ti sp a es { ( F γ , δ γ , ∆ γ ) } 0 <γ ≤ 1 suh that, for e ah 0 < γ ≤ 1 in K , ( F γ , δ γ , ∆ γ ) is a γ - ontr ation of ( F, δ, ∆) . F or every α, β ∈ K ∗ + suh that 0 < β ≤ α ≤ 1 , let f β ,α b e the transition b etwe en the two  ontr ations ( F α , δ α , ∆ α ) and ( F β , δ β , ∆ β ) of ( F, δ, ∆) . Then, in the  ate gory G K , the family { f β ,α } 0 <β <α ≤ 1 has a dir e t limit whih is a trivial galati sp a e. Pr o of. F rom the preeding prop osition, w e kno w that the pro dut of t w o transi- tions is a transition. No w, w e  ho ose a trivial galati spae ( F 0 , δ 0 , ∆ 0 ) and for Con tration of a Generalized Metri Struture 13 ea h 0 < γ ≤ 1 in K , let f 0 ,γ : ( F γ , δ γ , ∆ γ ) → ( F 0 , δ 0 , ∆ 0 ) b e a trivial morphism. Then, for an y α, β ∈ K su h that 0 < β ≤ α ≤ 1 , w e ha v e f 0 ,α = f 0 ,β ◦ f β ,α . No w, w e onsider a galati spae ( G, d, D ) and w e supp ose that, for ea h α ∈ K su h that 0 < α ≤ 1 and for ea h α -on tration ( F α , δ α , ∆ α ) of ( F, δ, ∆) , w e ha v e a morphism g α : ( F α , δ α , ∆ α ) → ( G, d, D ) su h that g α = g β ◦ f β ,α for ev ery elemen t β of K su h that 0 < β ≤ α and for ev ery β -on tration ( F β , δ β , ∆ β ) of ( F, δ, ∆) with transition f β ,α . Giv en t w o p oin ts x 1 and x 2 in F α w e an nd a suien tly small β 0 < α su h that δ β ( f β ,α ( x 1 ) , f β ,α ( x 2 )) = 0 for ev ery β ≤ β 0 . Hene, w e see that there is a single elemen t y 0 ∈ G su h that g α ( F α ) = { y 0 } for ea h α . Consequen tly , the onstan t map g 0 : F 0 → G with v alue y 0 is the unique morphism su h that g α = g 0 ◦ f 0 ,α for ea h α . 4.6 In the follo wing result, for ea h galati spae ( F, δ, ∆) , the set F is pro vided with the top ology dened b y the generalized distane δ and the set M F of its metri omp onen ts is pro vided with the top ology dened b y the galati distane ∆ . Prop osition 5. L et us  onsider a limite d element γ > 0 in K and a γ -  ontr ation f γ : ( F, δ, ∆) → ( F γ , δ γ , ∆ γ ) . 1. The maps f γ : F → F γ and e f γ : M F → M F γ ar e  ontinuous. 2. If γ 6≃ 0 , f γ and e f γ ar e home omorphisms. 3. If γ ≃ 0 , then for every z ∈ F γ , f γ − 1 ( { z } ) is an op en set of F . 4. If γ ≃ 0 and if we  an nd η ∈ K ∗ + suh that η ≃ 0 and γ /η ≃ 0 , then, for every Z ∈ M F γ the set e f − 1 γ ( { Z } ) is an op en set of M F . Pr o of. 1. Let V b e an op en set of F γ and z ∈ ( f γ ) − 1 ( V ) . Let z ′ ∈ V su h that f γ ( z ) = z ′ and r ∈ R ∗ + su h that the op en ball B δ γ ( z ′ , r ) is a subset of V . Then, from the ondition γ ≤ 1 w e dedue that B δ ( z , r ) ⊂ ( f γ ) − 1 ( B δ γ ( z ′ , r )) . Hene, the set f γ − 1 ( V ) is an op en set of F γ . Th us f γ is on tin uous. A similar argumen t sho ws that e f γ is also on tin uous. 2. W e supp ose that γ 6≃ 0 . Then, w e kno w that f γ is in v ertible and, for ev ery x ′ , y ′ ∈ F γ δ ( f − 1 γ ( x ′ ) , f − 1 γ ( y ′ )) = p v ( γ ) − 1 δ γ ( x ′ , y ′ ) ∆( C F ( f − 1 γ ( x ′ )) , C F ( f − 1 γ ( y ′ ))) = γ − 1 • ∆ γ ( C F γ ( x ′ ) , C F γ ( y ′ )) F rom this, w e dedue that f − 1 γ and ( e f γ ) − 1 are on tin uous. 3. W e supp ose no w that γ ≃ 0 . Giv en z ′ ∈ F γ , w e onsider an y z ∈ F su h that f γ ( z ) = z ′ . Let E ∈ M F su h that z ∈ E and let t an arbitrary p oin t of E . Sine δ ( z , t ) is limited w e see that δ γ ( f γ ( z ) , f γ ( t )) = 0 . Th us E ⊂ f − 1 γ ( { z ′ } ) and sine E is a neigh b orho o d of z in F , w e get that f − 1 γ ( { z ′ } )) is op en. 4. W e supp ose that γ ≃ 0 and that w e an nd η ∈ K ∗ + su h that η ≃ 0 and γ /η ≃ 0 . W e onsider Z ∈ M F γ and let E ∈ M F su h that e f γ ( E ) = Z . Then, ρ = Gal ( η − 1 ) is stritly greater than 0 := Gal (0) in the ordered group Con tration of a Generalized Metri Struture 14 Gal ( K ) . W e onsider an elemen t H of the 'op en ball' B ∆ ( E , ρ ) of M F . Th us, ∆( E , H ) < ρ ; onsequen tly , H ∈ e f − 1 γ ( { Z } sine ∆ γ ( e f γ ( E ) , e f γ ( H )) = 0 b eause γ • ∆( E , H ) ≤ γ • ρ = Gal ( γ /η ) = Gal (0) = 0 . Hene, B ∆ ( E , ρ ) ⊂ e f − 1 γ ( { Z } ) and this last set is op en. If γ ≃ 0 , w e notie that the prop ert y ∃ η ∈ K ∗ + tel que η ≃ 0 et γ /η ≃ 0 is not satised in ev ery ordered eld extension of R : for instane, if w e  ho ose γ := X , w e annot nd su h a η in the eld or rational funtions R ( X ) or in the eld or Lauren t series R (( X )) . In the eld R [[ X Q ]] of Puiseux series or in a eld ∗ R of h yp erreal n um b ers, this prop ert y is true for ev ery γ ≃ 0 . 5 Nonstandard saling of a metri spae In all this setion, w e onsider a metri spae ( X, d ) . W e need some nonstandard extensions of R , X and d . T o this end, w e an use the metho d of ultra-p o w ers as in setion 2. More generally [8, 14 ℄, w e an onsider a sup erstruture V ( S ) o v er a set S su h that ( X ∪ R ) ⊂ S and a nonstandard mo del of V ( S ) V ( S ) − → V ( ∗ S ) Y 7− → ∗ Y with a large enough saturation prop ert y (our study do es not require an y parti- ular renemen t in the  hoie of the nonstandard mo del). Equiv alen tly , w e an used the axiomati approa h of Hrba£ek [9℄. Notie that Nelson's in ternal set theory IST [15 ℄ is not adapted to this w ork sine it do es not allo w a on v enien t treatmen t of external sets. Then w e get at the same time the nonstandard extensions ∗ R or R , ∗ X of X and ∗ d : ∗ X × ∗ X → ∗ R + of d pro vided b y the giv en nonstandard mo del. 5.1 Ea h elemen t of the m ultipliativ e group ∗ R ∗ + := { γ ∈ ∗ R ; 0 < γ } of stritly p ositiv e h yp erreal n um b ers is alled a s ale . Giv en a sale α ∈ ∗ R ∗ + , w e dene the equiv alene relation ≃ α on ∗ X dened b y ∀ ( x, y ) ∈ ∗ X 2 ( x ≃ α y ⇐ ⇒ α ∗ d ( x, y ) ≃ 0) Then, w e in tro due the quotien t set X α = ∗ X / ≃ α and the anonial pro jetion π α : ∗ X − → X α x 7− → π α ( x ) where π α ( x ) denotes the equiv alene lass of x ∈ ∗ X , i.e the set of y ∈ ∗ X su h that x ≃ α y . Asso iated to the h yp er-distane ∗ d , there is a natural map δ α : X α × X α − → R + ∪ { + ∞} ( ξ , η ) 7− → δ α ( ξ , η ) Con tration of a Generalized Metri Struture 15 su h that, if ξ = π α ( x ) and η = π α ( y ) , then δ α ( ξ , η ) = st ( α ∗ d ( x, y )) . It is lear that δ α is a generalized distane on X α . Let M α b e the set of metri omp onen ts of X α for δ α . Th us ea h E ∈ M α is of the follo wing form E = Cone ( X, x E , α ) := { x ∈ ∗ X ; α ∗ d ( x, x E ) 6≃ + ∞} / ≃ α where x E is an y p oin t in ∗ X su h that π α ( x E ) ∈ E . When α ≃ 0 , the set Cone ( X, x E , α ) is exatly the so-alled asymptoti  one of ( X, d ) with resp et to x E and α . W e reall that Gal is the anonial pro jetion ∗ R → Gal ( ∗ R ) = ∗ R / Gal (0) . F or ev ery E 1 and E 2 in M α , w e  ho ose x E 1 and x E 2 in ∗ X su h that E 1 = Cone ( X, x E 1 , α ) and E 2 = Cone ( X, x E 2 , α ) ; then w e dene ∆ α ( E 1 , E 2 ) = Gal ( α ∗ d ( x E 1 , x E 2 )) Th us, w e get a map ∆ α : M 2 α → Gal ( ∗ R ) whi h is a galati distane and w e an onsider the galati spae ( X α , δ α , ∆ α ) . Denition 7. Given a s ale α ∈ ∗ R ∗ + , the (nonstandar d) α -saling of the metri sp a e ( X, d ) is the galati sp a e ( X α , δ α , ∆ α ) . Hene, starting from a usual metri spae ( X, d ) , w e get a family ( X α , δ α , ∆ α ) α ∈ ∗ R ∗ + of galati spaes whi h are the dieren t saling of ( X, d ) . 5.2 Let us no w onsider t w o sales α, β ∈ ∗ R ∗ + su h that β < α . No w, w e w an t to ompare the α -saling and the β -saling of our metri spae ( X, d ) . If ( x, y ) ∈ ∗ X 2 is su h that α ∗ d ( x, y ) ≃ 0 , then β ∗ d ( x, y ) ≃ 0 . Therefore, there exists a natural surjetiv e map π β ,α : X α → X β su h that π β = π β ,α ◦ π α . In the same w a y , if γ ∈ ∗ R ∗ + is su h that γ ≤ β ≤ α , then π γ ,α = π γ ,β ◦ π β ,α . Theorem 3. The map π β ,α is a ( β /α ) - ontr ation fr om the α -s aling ( X α , δ α , ∆ α ) of ( X, d ) onto its β -s aling ( X β , δ β , ∆ β ) . In other w ords, the β -saling of ( X, d ) is a ( β /α ) -on tration of its α -saling. Consequen tly , insofar as w e are only onerned b y the struture of galati spae, w e an dene the β -saling of ( X, d ) using only its α -saling. Pr o of. It is lear that π β ,α is a surjetiv e map from X α to X β . F urthermore, if ξ = π α ( x ) and η = π α ( y ) , then δ α ( ξ , η ) = st ( α ∗ d ( x, y )) and δ β ( π β ,α ( ξ ) , π β ,α ( η )) = st ( β ∗ d ( x, y )) Therefore δ β ( π β ,α ( ξ ) , π β ,α ( η )) = st (( β /α ) α ∗ d ( x, y )) hene δ β ( π β ,α ( ξ ) , π β ,α ( η )) =  st (( β /α ) δ α ( ξ , η )) if δ α ( ξ , η ) < + ∞ st ( γ . ∆( C X α ( ξ ) , C X α ( η ))) if δ α ( ξ , η ) = + ∞ . In the same w a y , if E = Cone ( X, x, α ) and F = Cone ( X, y , α ) , then ∆ β ( π β ,α [ E ] , π β ,α [ F ]) = Gal ( β ∗ d ( x, y )) = ( β / α ) • ∆ α ( E , F ) Con tration of a Generalized Metri Struture 16 F rom this, it results that the maps π β ,α ha v e all the prop erties of transitions stated in the preeding setion. F or instane, the limit of ( X α , δ α , ∆ α ) when α → 0 in K ∗ + is a trivial galati spae. 5.3 A new feature ab out the family of nonstandard salings of a metri spae ( X, d ) is that the galati spaes ( X α , δ α , ∆ α ) are dened for arbitrary large sale α ∈ ∗ R ∗ + . Then, a natural question is related to the existene of the limit of ( X α , δ α , ∆ α ) when α → + ∞ in ∗ R ∗ + . Let us all a hain an y family ξ = ( ξ α ) α ∈ ∗ R ∗ + su h that, for ea h α ∈ ∗ R ∗ + the elemen t ξ α b elongs to X α and π β ,α ( ξ α ) = ξ β for ea h β ≤ α ∈ ∗ R ∗ + . Prop osition 6. L et us  onsider two hains ξ = ( ξ α ) α ∈ ∗ R ∗ + and ξ ′ = ( ξ ′ α ) α ∈ ∗ R ∗ + suh that ξ 6 = ξ ′ . F or every α ∈ ∗ R ∗ + , let E α and E ′ α b e the metri  omp onents of r esp e tively ξ α and ξ ′ α in X α . Then, ther e is α 0 ∈ ∗ R ∗ + suh that, for every α ≥ α 0 , we have δ α ( ξ α , ξ ′ α ) = + ∞ . F urthermor e, lim α → + ∞ ∆ α ( E α , E ′ α ) = + ∞ for the or der top olo gy on ∗ R ∗ + and Gal ( ∗ R ) . Pr o of. F or ea h α ∈ ∗ R ∗ + , w e  ho ose x α , x ′ α ∈ ∗ X su h that π α ( x α ) = ξ α and π α ( x ′ α ) = ξ ′ α . Sine ξ 6 = ξ ′ , there is β ∈ ∗ R ∗ + su h that ξ β 6 = ξ ′ β . Th us β ∗ d ( x β , x ′ β ) 6≃ 0 . Let α a sale su h that α > β ; sine π β ( x α ) = π β ,α ◦ π α ( x α ) = π β ( x β ) and π β ( x ′ α ) = π β ,α ◦ π α ( x ′ α ) = π β ( x ′ β ) , w e ha v e β ∗ d ( x α , x β ) ≃ 0 and β ∗ d ( x ′ α , x ′ β ) ≃ 0 and th us β ∗ d ( x α , x ′ α ) 6≃ 0 . Hene, if w e  ho ose α 0 su h that α 0 /β ≃ + ∞ , w e ha v e for ev ery α ≥ α 0 δ α ( ξ α , ξ ′ α ) = st ( α ∗ d ( x α , x ′ α )) = st (( α/β ) β ∗ d ( x α , x ′ α )) = + ∞ Moreo v er, sine lim α → + ∞ ∆ α ( E α , E ′ α ) = Gal ( α ∗ d ( x α , x ′ α )) = Gal (( α/β ) β ∗ d ( x α , x ′ α )) w e see that ∆ α ( E α , E ′ α ) on v erges to w ards + ∞ in Gal ( ∗ R ) when α → + ∞ in ∗ R ∗ + . The last result suggests that, if w e w an t to nd a limit for ( X α , δ α , ∆ α ) when α → + ∞ , w e ha v e to widen the ategory G ∗ R of galati spaes. T o this end, w e in tro due the ategory G ′ ∗ R of gener alize d galati sp a es . The ob jets of this ategory are strutures ( X, δ, ∆) where X is a set, δ is a generalized distane on X and ∆ is a gener alize d galati distan e on the set M X of metri omp onen ts of X for δ . This last ondition means that ∆ is a map M X × M X → Gal ( ∗ R ) ∗ + ∪ { + ∞} whi h satises the general metri rule. In G ′ ∗ R , a morphism from ( X, δ, ∆) to ( X ′ , δ ′ , ∆ ′ ) is a map f : X → X ′ su h that there is a limited elemen t γ > 0 in K so that the follo wing onditions are satised 1. ∀ ( x 1 , x 2 ) ∈ X 2 δ ′ ( f ( x 1 ) , f ( x 2 )) ≤ γ δ ( x 1 , x 2 ) (th us, f indues a map e f from the set M X of metri omp onen ts of X for δ to the set M X ′ of metri omp onen ts of X ′ for δ ′ ); 2. ∀ ( E 1 , E 2 ) ∈ M X 2 ∆ ′ ( e f ( E 1 ) , e f ( E 2 )) ≤ γ • ∆( E 1 , E 2 ) . W e denote b y lim α → + ∞ ( X α , δ α , ∆ α ) the in v erse (or pro jetiv e) limit of the fam- ily ( π β ,α ) β ≤ α ∈ ∗ R ∗ + of morphisms π β ,α : ( X α , δ α , ∆ α ) → ( X β , δ β , ∆ β ) in the ategory G ′ ∗ R . If this limit exists, it is w ell dened up to an isomorphism. Con tration of a Generalized Metri Struture 17 Prop osition 7. F or e ah metri sp a e ( X, d ) , the limit lim α → + ∞ ( X α , δ α , ∆ α ) ex- ists and is e qual to the p air  ( X ∞ , δ ∞ , ∆ ∞ ) , ( π α, ∞ ) α ∈ ∗ R +  wher e ( X ∞ , δ ∞ , ∆ ∞ ) is a gener alize d galati sp a e and ( π α, ∞ ) α ∈ ∗ R + is a family of morphisms fr om ( X ∞ , δ ∞ , ∆ ∞ ) to ( X α , δ α , ∆ α ) suh that • X ∞ is the set of hains (families ξ = ( ξ α ) α ∈ ∗ R ∗ + suh that ξ α ∈ X α for al l α ∈ ∗ R + and π β ,α ( ξ α ) = ξ β for al l β ≤ α ∈ ∗ R + ). • ∀ ( ξ , η ) ∈ X 2 ∞ δ ∞ ( ξ , η ) =  0 if ξ = η + ∞ if ξ 6 = η • ∀ ( ξ , η ) ∈ X 2 ∞ ∆ ∞ ( { ξ } , { η } ) =  0 if ξ = η + ∞ if ξ 6 = η • ∀ ξ = ( ξ α ) α ∈ ∗ R ∗ + ∈ X ∞ ∀ α ∈ ∗ R ∗ + π α, ∞ ( ξ ) = ξ α This result sa ys that the limit of the α -saling of ( X, d ) when α approa hes + ∞ is a h uge generalized galati spae in whi h the distane b et w een t w o dieren t p oin ts is + ∞ and the galati distane b et w een t w o dieren t metri omp onen ts is + ∞ . If ∗ X denotes the nonstandard extension of X used in the onstrution of the saling of ( X, d ) , w e see that there is a natural map ∗ X − → X ∞ x 7− → ( π α ( x )) α ∈ ∗ R ∗ + whi h is learly injetiv e but w e do not kno w if it is surjetiv e. Pr o of of the pr op osition. W e remark that the set of metri omp onen ts of X ∞ is M X ∞ = { { ξ } ; ξ ∈ X ∞ } It is lear that ea h π α, ∞ is a map X ∞ → X α su h that π β , ∞ = π β ,α ◦ π α, ∞ whenev er β ≤ α ∈ ∗ R + . In an ob vious w a y , π α, ∞ is a morphism of gener- alized galati spae (with a Lips hitz onstan t equal to 1 for instane) from ( X ∞ , δ ∞ , ∆ ∞ ) to ( X α , δ α , ∆ α ) . Let ( Y , d, D ) b e a generalized galati spae and ( ψ α ) α ∈ ∗ R + b e a family of mor- phisms ψ α from ( Y , d, D ) to ( X α , δ α , ∆ α ) su h that ψ β = π β ,α ◦ ψ β for all β ≤ α ∈ ∗ R ∗ + . F or ea h y ∈ Y , the family ψ ∞ ( y ) = ( ψ α ( y )) α ∈ ∗ R + b elongs to X ∞ . Th us, w e get a map ψ ∞ : Y → X ∞ whi h is the unique map whi h satises ψ α = π α, ∞ ◦ ψ ∞ for all α ∈ ∗ R + . Let y and y ′ t w o p oin ts of Y su h that ψ ∞ ( y ) 6 = ψ ∞ ( y ′ ) . Hene, there is α 0 ∈ ∗ R + su h that ψ α ( y ) 6 = ψ α ( y ′ ) for ev ery α ≥ α 0 . F or suien tly large α , w e see that δ α ( ψ α ( y ) , ψ α ( y ′ ) = + ∞ and th us d ( y , y ′ ) = + ∞ . Consequen tly , ψ ∞ indues a map e ψ ∞ : M Y → M X ∞ . Let E , E ′ ∈ M Y su h that e ψ ∞ ( E ) 6 = e ψ ∞ ( E ′ ) . There are ξ 6 = ξ ′ ∈ X ∞ su h that e ψ ∞ ( E ) = { ξ } and e ψ ∞ ( E ′ ) = { ξ ′ } . F or ea h α ∈ ∗ R ∗ + , there is a real n um b er k α > 0 su h that ∆ α ( e ψ α ( E ) , e ψ α ( E ′ )) ≤ k α D ( E , E ′ ) and ∆ α ( e ψ α ( E ) , e ψ α ( E ′ )) = Gal ( α ∗ d ( x α , x ′ α )) where x α , x ′ α are elemen ts of ∗ X su h that π α ( x α ) = π α, ∞ ( ξ ) and π α ( x ′ α ) = π α, ∞ ( ξ ′ ) . Sine Gal ( α ∗ d ( x α , x ′ α )) is arbitrarily large in the ordered group Gal ( ∗ R ) when α → + ∞ and sine k α is a real n um b er, w e dedue that D ( E , E ′ ) = + ∞ . Th us, ψ ∞ is a morphism ( Y , d, D ) → ( X ∞ , δ ∞ , ∆ ∞ ) in the ategory of generalized galati spaes. Con tration of a Generalized Metri Struture 18 5.4 W e ma y think there is a link b et w een the onept of Gromo v-Hausdor on v ergene and our notion on nonstandard saling of a metri spae. Firstly , let us reall what is the Gromo v-Hausdor distane [ 6℄. Giv en t w o subsets A and B of a metri spae ( Z, δ ) , the Hausdor distane b et w een A and B in Z is d Z H ( A, B ) := inf { ε ∈ R ∗ + ; A ⊂ V ε ( B ) and B ⊂ V ε ( A ) } where, for ea h subset C ⊂ Z , the set V ε ( C ) is the ε -neigh b ourho o d { x ∈ Z ; δ ( x , C ) < ε } of C . Then, the Gr omov-Hausdor distan e d GH ( E , F ) of t w o metri spaes E and F is the inm um of n um b ers d Z H ( i ( E ) , j ( F )) for an y ( Z, i, j ) su h that Z is a metri spae, i : E → Z and j : F → Z are isometri em b eddings. The Gromo v-Hausdor distane is not really a distane, mainly b eause there are non isometri metri spaes E and F su h that d GH ( E , F ) = 0 (for instane R and Q ). This problem disapp ears in the olletion of isometri lasses of ompat metri spaes. Nev ertheless, w e sa y that a sequene ( E n ) of metri spaes (ompat or not ompat) on v erges to w ard a metri spae F for the Gromo v-Hausdor distane if d GH ( E n , F ) on v erges to 0 in R + when n → + ∞ . Theorem 4. L et ( λ n ) b e a standar d se quen e of stritly p ositive r e al numb ers suh that lim n → + ∞ λ n = 0 . If the se quen e ( X, λ n d )  onver ges to a metri sp a e ( F, d F ) for the Gr omov-Hausdor distan e, then, for e ah innitely lar ge ν ∈ ∗ N , the λ ν -s aling X λ ν of X is a galati sp a e isometri to the 1 -s aling F 1 of ( F, d F ) . Pr o of. Firstly , w e giv e a more on v enien t form ulation of the Gromo v-Hausdor distane d GH ( E , F ) of t w o metri spaes ( E , d E ) and ( F, d F ) : it is the inm um of the set of real n um b ers ε > 0 su h that there exists a map δ : E × F → R +  he king the t w o follo wing prop erties: 1. the map d : ( E ∐ F ) 2 → R + dened b y d ( x, y ) =        d E ( x, y ) if ( x, y ) ∈ E 2 d F ( x, y ) if ( x, y ) ∈ F 2 δ ( x, y ) if ( x, y ) ∈ E × F δ ( y , x ) if ( x, y ) ∈ F × E is su h that d ( x, z ) ≤ d ( x, y ) + d ( y , z ) for ev ery x , y and z in the disjoin t union E ∐ F of E and F , 2. ( ∀ x ∈ E ∃ y ∈ F δ ( x, y ) < ε ) and ( ∀ x ∈ F ∃ y ∈ E δ ( y, x ) < ε ) . No w w e return to the pro of of the theorem. F rom the on v ergene h yp othesis, w e dedue that, there is a sequene ( ε n ) of real n um b ers su h that l im n → + ∞ ε n = 0 and there is a sequene ( δ n ) of maps from X × F to R + su h that, for all n ∈ N : 1. the map d n : ( X ∐ F ) 2 → R + dened b y d n ( x, y ) =        λ n d ( x, y ) si ( x, y ) ∈ X 2 d F ( x, y ) si ( x, y ) ∈ F 2 δ n ( x, y ) si ( x, y ) ∈ X × F δ n ( y , x ) si ( x, y ) ∈ F × X is su h that d n ( x, z ) ≤ d n ( x, y ) + d n ( y , z ) for ev ery x , y and z in the disjoin t union X ∐ F of X and F , Con tration of a Generalized Metri Struture 19 2. ( ∀ x ∈ X ∃ y ∈ F δ n ( x, y ) < ε n ) and ( ∀ x ∈ F ∃ y ∈ X δ n ( y , x ) < ε n ). Let ν b e an innitely large elemen t of ∗ N . Th us, w e get a map δ ν : ∗ X × ∗ F → ∗ R and a map d ν : ( ∗ X ∐ ∗ F ) 2 → ∗ R su h that 1. ∀ ( x, y ) ∈ ( ∗ X ∐ ∗ F ) 2 d ν ( x, y ) =        λ ν d ( x, y ) si ( x, y ) ∈ ∗ X 2 d F ( x, y ) si ( x, y ) ∈ ∗ F 2 δ ν ( x, y ) si ( x, y ) ∈ ∗ X × ∗ F δ ν ( y , x ) si ( x, y ) ∈ ∗ F × ∗ X 2. ∀ x, y , z ∈ ∗ X ∐ ∗ F d ν ( x, z ) ≤ d ν ( x, y ) + d ν ( y , z ) , 3. ( ∀ x ∈ ∗ X ∃ y ∈ ∗ F δ ν ( x, y ) ≃ 0) and ( ∀ x ∈ ∗ F ∃ y ∈ ∗ X δ ν ( y , x ) ≃ 0) . Then, w e onsider the quotien t set G := ∗ X ∐ ∗ F / ∼ for the equiv alene relation ∀ x, y ∈ ∗ X ∐ ∗ F ( x ∼ y ⇐ ⇒ d ν ( x, y ) ≃ 0) . 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