No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit

No em b edding of the automorphisms of a top ological space in to a compact metric space endo ws them with a comp ositio n that passes to the limit P atrizio F rosini Dip artimento di Matematic a, Universit` a di Bolo gna Claudia Landi Dip artimento di Scienze e Meto di del l’Inge gneria, Unive rsit` a di Mo dena e R e ggio Emilia Abstract The Hausdorff distance, the Gromo v-Hausdorff, the F r ´ ec het and the natu- ral pseudo-distances are instance s of dissimilarit y measures widely used in shap e c omparison. W e sho w that they share the prop ert y of b eing defined as inf ρ F ( ρ ) where F is a suitable functional and ρ v aries in a set of corresp on- dences con taining the set of homeomorphisms. Our ma in result states that the set of homeomorphisms cannot b e enlarged to a metric space K , in suc h a w ay tha t the comp osition in K (extendin g the comp osition of homeomor- phisms) passes to the limit and, at the same time, K is compact. Keywor ds: Space of homeomorphisms, corresp ondence, compact metric space 2010 MSC: Primary 57S05, 5 7S10; Secondary 54C35, 6 8 U05 1. In tr oduction The literature a bout shap e comparison o ften reports distances or pseudo- distances whose definitions are based on considering sets of corresp ondences b et w een topo lo gical spaces X and Y , where a corresp ondence is defined as a surjectiv e relation ρ ⊆ X × Y suc h that also ρ − 1 is surjec tive ( M´ e moli , 2007). In plain w ords, eac h corresp ondence des crib es a (p erceptiv e) m at c hing b et w een the p oin ts of X and the p oin ts o f Y . As a classical example, t he Hausdorff distanc e d H ( X , Y ) betw een t w o non-empt y compact sets X and Y of a metric space ( S , d S ) is defined as the Pr eprint submitte d to Applie d Mathematics L ett ers Novemb er 14, 2018 v alue inf ρ ∈C sup ( x,y ) ∈ ρ d S ( x, y ), where C denotes the set of all cor r es p ondences b et w een X and Y (M´ emoli, 2007). The Gr omov-Hausdorff pse udo- distanc e (Burago, Burago and Iv ano v, 200 1 ; G r o mo v, 1981) and the F r´ echet pseudo- distanc e (Rote, 2007) represen t t wo o ther w ell-know n examples where a sim- ilar pro cedure is applied. Sometimes (as in the case of t he F r ´ ec het pseudo- distance) just a pro per subset o f the set of all corresp ondenc es is considered. All these ex amples share the prop ert y of b eing defined as inf ρ F ( ρ ), where F is a suitable functional taking eac h corresp ondence ρ to a v alue tha t mea- sures how m uc h “ ρ b eha v es as an iden tit y” from the p oin t of view of our shap e comparison. In the case of the Hausdorff distance, F ( ρ ) equals the v alue s up ( x,y ) ∈ ρ d S ( x, y ), whic h v anishes if and only if X = Y and ρ is the iden tit y corresp ondence. In Pe rsisten t T op ology the same pro ced ure leads to the concept o f nat- ur al pseudo-distanc e , considering only corresp ondences that are also home- omorphisms. Whe n t w o closed C 0 manifolds X , Y endo w ed with t w o con- tin uous functions ϕ : X → R , ψ : Y → R are considere d together with the set H om ( X , Y ) of all homeomorphisms b et w een X and Y , this extended pseudo-distance is defined to b e either the v alue inf h ∈ H om ( X, Y ) max x ∈ X | ϕ ( x ) − ψ ( h ( x )) | , or + ∞ , dep endin g on whether X and Y are homeomorphic or not (F rosini and Mulazzani, 1999; D onatini a nd F r o sini, 200 4 , 2007, 2 009). W e observ e that the sets of corresp ondenc es considered in our examples include all homeomorphisms, whic h are alw ays assumed to b e legitimate transformations. Unfortunately , in all the previous examples at least one of the follow ing problems o ccurs: (1) the comp osition of relations do es not pass to the limit; (2) the infim um of the functional F is not a minimum. Consequen tly , a natural goa l w ould b e to guaran tee tha t our functional a ttains a minim um b y extending the metric space of the homeomorphisms b et w een tw o any top ological spaces X and Y to a compact metric space whos e elemen ts are (p ossibly but not necessarily) corresp ondences , endow ed with a comp osition that extends the us ual comp osition of homeomorphisms a nd passes to the limit. The purpose of this pap er is proving that this goal cannot be reac hed ev en in the case X = Y , under prett y reasonable h yp otheses. This fact suggests the e xistence of obs ta cle s in treating, exclusiv ely in terms of corres p ondences, the distances defined as inf ρ F ( ρ ). 2 2. General sett ing Let us denote by C any small category (i.e. an y category C suc h that b oth O bj ( C ) and M or ( C ) are actually sets) ha ving the following pro perties: 1. its ob jects are top ological spaces; 2. each (p ossibly empt y) set of morphisms M or ( X , Y ) b et w een tw o ob- jects X and Y is a subs et of the set o f corresp ondences from X onto Y , con taining all the p ossible ho meomorphisms fro m X on to Y ; 3. if ρ ∈ M or ( X , Y ) then ρ − 1 ∈ M or ( Y , X ). V arying ( X , Y ) in the set O bj ( C ) × O bj ( C ), let us consider a family o f functionals F ( X,Y ) : M or ( X, Y ) → R satisfying t he following prop erties: 1. fo r ev ery ρ ∈ M or ( X , Y ), F ( X,Y ) ( ρ ) ≥ 0; 2. if id X is the identit y morphism on X , then F ( X,X ) ( id X ) = 0 ; 3. fo r ev ery ρ ∈ M or ( X , Y ), F ( X,Y ) ( ρ ) = F ( Y ,X ) ( ρ − 1 ); 4. if ρ ∈ M or ( X, Y ) and σ ∈ M or ( Y , Z ), F ( X,Z ) ( σ ◦ ρ ) ≤ F ( X,Y ) ( ρ ) + F ( Y ,Z ) ( σ ). The family of functionals F ( X,Y ) allo ws us to define an extended pseudo- distance on O bj ( C ) ( we omit the trivial pro of ). The term extende d means that the pseudo-distance can take the v a lue + ∞ . Ob viously , passing t o the quotien t, an y pseudo-distance b ecomes a distance (i.e. also the axiom d ( X , Y ) = 0 = ⇒ X = Y is satisfied). Prop osition 2.1. T he function δ ( X , Y ) =  inf ρ ∈ M or ( X , Y ) F ( X,Y ) ( ρ ) if M or ( X , Y ) 6 = ∅ , + ∞ if M or ( X, Y ) = ∅ is an extende d pseudo-dis tanc e on O bj ( C ) . The previous setting allo ws us t o obtain the pseudo-distances w e ha v e recalled at the b eginning of the in tro duction, a s particular cases. Hausdorff distance . C is the category whose ob jects are the non-empt y compact subse ts of a metric space ( S , d S ). The morphisms are all corresp on- dences b et w een an y t w o ob jects. W e set F ( X,Y ) ( ρ ) = sup ( x,y ) ∈ ρ d S ( x, y ), for ev ery pair ( X , Y ) ∈ O bj ( C ) × O bj ( C ) and ev ery ρ ∈ M o r ( X , Y ). 3 Gromo v-Hausdorff pseu do-distance . C is a category whose ob jec ts b e- long to a set of non-empt y compact metric spaces. The mor phis ms are giv en b y a ll c orr es p ondences betw een ob j ects. F o r ev ery ( X, Y ) ∈ O bj ( C ) × O bj ( C ) and ρ ∈ M or ( X, Y ), w e set F ( X,Y ) ( ρ ) = inf ( Z ,d Z ) ,f ,g sup ( x,y ) ∈ ρ d Z ( f ( x ) , g ( y )), where ( Z , d Z ) ranges ov er all metric spaces, and f and g ra ng e ov er all p os- sible isometric embeddings of X a nd Y in to Z , resp ectiv ely . F r´ ec het pseudo-distance . C is the category whose ob jec ts a re all the curv es γ : [0 , 1] → R n (seen as subsets of [0 , 1] × R n endo w ed with the pro duct top ology). The morphisms betw een t w o curv es γ 1 , γ 2 are given b y the relations ρ whose elemen ts can b e written a s ( γ 1 ( α ( t )) , γ 2 ( β ( t ))), where t ∈ [0 , 1] and α, β : [0 , 1] → [0 , 1] are tw o non- dec reasing and surjectiv e con- tin uous f unctions. Finally , w e set F ( X,Y ) ( ρ ) = sup ( x,y ) ∈ ρ k x − y k . Natural pseudo-d ist ance . C is the category whose ob jects are all the con tin uous functions ϕ : X → R , where X ranges ov er all closed C 0 n - manifolds. They are seen as subs ets o f X × R , endo w ed with the pro duct top ology . The morphisms b et w een tw o f unc tio ns ϕ : X → R , ψ : Y → R are giv en b y t he homeomorphisms h f r o m X on to Y . Finally , w e set F ( X,Y ) ( h ) = max x ∈ X | ϕ ( x ) − ψ ( h ( x )) | . 3. Main result The core of this pap er is the following result stating that we cannot enlarge the set of homeomorphisms to a larger metric space K , in suc h a w ay that the comp osition in K (extending the comp osition of homeomorphisms) passes to the limit and, at the same time, K is compact. Since the passage to the limit of the comp osition is imp ortan t in applications b ecause of the need for computatio nal approx imat io ns , our result suggests that there is no sensible w a y to extend the set of homeomorphis ms to a larger compact me tric space. Theorem 3.1. L et X b e a top olo gic al sp ac e c ontaining a subset U that is home omorphic to an n -dime n sional op en b al l for some n ≥ 1 . L et us c onsider the se t H of a l l home omorph isms fr om X onto X , endowe d with a metric d H that is c omp atible with the top olo gy of X in the sense of the fol low i n g pr op erty: if a se quenc e ( h i ) i n H c onve r ges to the iden tic al h ome om orphism id X ∈ H with r esp e ct to d H , then ( h i ) p ointwise c onver ges to id X with r esp e c t 4 to the top olo gy of X (i.e., lim i →∞ h i ( x ) = x for every x ∈ X ). Then no c omp act metric sp ac e ( K , d K ) exists, endowe d with an internal c omp osition • : K × K → K such that: 1. K ⊇ H ; 2. d K extends d H (i.e. if f , g ∈ H then d K ( f , g ) = d H ( f , g ) ) ; 3. the binary op er ation • extends the usual c omp osition of home omor- phisms (i.e., if f , g ∈ H then f • g = f ◦ g ); 4. the c om p osition • c ommutes with the p a ssage to the limit (i.e . if the se quenc es ( ρ i ) and ( σ i ) c onver ge in K , then lim i →∞ ( ρ i • σ i ) exists and e quals (lim i →∞ ρ i ) • (lim i →∞ σ i ) ). Pr o of. Let us pro ve our result by contradiction, assuming that suc h a met- ric space ( K , d K ) exists. F o r ev ery ho me o mo r phis m f ∈ H and any natur a l n um b er i > 1, let f i denote the composition of f with itself i times (while f 1 = f ), and let us set g = f − 1 . Since K is compact, a strictly increasing se- quence of p ositiv e n um b ers ( i r ) exists suc h that b oth the limits, with resp ect to d K , lim r →∞ f i r and lim r →∞ g i r exist. On one hand, if in the metric space ( K , d K ) w e consider the constan t sequence ( f i r ◦ g i r ) = ( id X ), from Prop erties 3 and 4 it follows tha t id X = lim r →∞  f i r ◦ g i r  = lim r →∞  f i r • g i r  =  lim r →∞ f i r  •  lim r →∞ g i r  =  lim r →∞ f i r +1  •  lim r →∞ g i r  = lim r →∞  f i r +1 • g i r  = lim r →∞  f i r +1 ◦ g i r  . Therefore, recalling Prop erties 1 and 2, w e ha v e that the sequence of homeomorphisms ( f i r +1 ◦ g i r ) = ( f i r +1 − i r ) conv erges to the iden tical home- omorphism, with respect to both d H and d K . W e observ e that eac h index i r +1 − i r is strictly p ositiv e. In other w ords, w e hav e pro ve d that for ev ery homeomorphism f from X on to X a sequenc e of po sitive num bers ( m r ) ex ists, suc h that ( f m r ) con v erges to the identic a l homeomorphism with resp ect to d H . In order to obtain a con tradiction, it is sufficien t to construct a homeomor- phism h that cannot ve rif y the previous prop ert y . W e can do that b y consid- ering a h o me o mo r phism ˜ h : U → B n = { x ∈ R n : k x k ≤ 1 } , and constructing a ho meomorphism h ∈ H that tak es the set ˜ h − 1  { x ∈ R n : k x k ≤ 1 2 }  in to the set ˜ h − 1  { x ∈ R n : k x k ≤ 1 4 }  . It is immediate to chec k tha t no sequence of non-trivial p ositiv e p o w ers of h can p oin t wise con v erge t o the identical homeomorphism. Therefore, no suc h a sequence can con ve rg e to the iden ti- cal homeomorphism with resp ect to d H . 5 R emark 3.2 . The assumption that X contains a subse t U that is homeomor- phic to an n -dimensional op en ball f o r s ome n ≥ 1 cannot b e omitted. Indeed, top ological spaces fo r whic h the only automorphism of X is the iden tity map exist. In that case ( H , d H ) is ob viously compact. A classical reference for these spaces (called rigid top olo gic al sp ac es ) is D e G roo t and Wille (1958). R emark 3.3 . An imp ortan t class of top ological spaces for whic h our t heo- rem holds is giv en b y the triangulable spaces (i.e. the b o dies of sim plicial complexes) of dimension larger than or equal to 1. Ac knowled gement. W e thank F rancesca Caglia r i for her v aluable sug- gestions. References D. Burago , Y. Burago , S. 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