D IAGRAMMES E LISABETH B URRONI In the heart of representable metric jets Diagrammes, tome S67-68 (2012), p. 33-52 © Université Paris 7, UER math., 2012, tous droits réservés. L’accès aux archives de la revue « Diagrammes » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ DIAGRAMMES , SUPPLEMENT AUX VOLUMES 67 + 68 , 2012, PARIS, pp. 33-52 IN THE HEART OF REPRESENTABLE METRIC JETS E. BURRONI EDITE PAR L. COPPEY , F. FOLTZ , R. GUITART, C. HENRY , C. LAIR 35 In the heart of representable metric jets Elisabeth Burroni This article is dedicated to Andrée Ehresmann When I rst met Andrée, she was 35, and a young dynamic mathematician. She is the one who read the six thesis (A. and E.Burroni, R.Guitart, Ch.Lair, M. and G.Weidenfeld) defended at Paris 7 in june 1970, Charles Ehresmann being our doctoral advisor. Since then, I have seen her every week in Paris along with Charles Ehresmann at Ehresmann's Seminar, as well as during the international Category Conferences that she used to organize at the University of Amiens where she was teaching. I remember the evenings at those Parisian restaurants where we were generously invited by Andrée and Charles Ehresmann along with visiting Categoricians ; that allowed us to meet foreign researchers in the eld of Category Theory. Finally, I am particularly grateful to Andrée for having believed in our joint work (Jacques Penon's and mine) about our metric Dierential Calculus. Here is a synthetic retrospective presentation of this work, skimming through our previous papers already published in arXiv [1], TAC [2], the Cahiers [3] and JPAA [6]. 1 Introduction Here, I aim to immerse myself in the heart of the metric jets, more precisely of those which are representable, restricting myself to the main basic concepts, while going deeper into some notions already mentionned in our previous papers ; this will give me the opportunity of lightening the previous texts (including some proofs), while precising some ideas and giving new examples (as the bifractal wave function) with a proof at the end of this paper. Concerning the concrete examples found all along this paper : they play the starring role in the understanding of our metric Dierential Calculus ! 36 I give a glossary at the end of this paper which briey recalls some useful notations and denitions (the rst occurrence of a new notion quoted in the text will be followed by a *, which suggests to refer to this glossary). 2 Metric jets So, mainly, as its name shows it, our metric Dierential Calculus generalizes the classical Dierential Calculus in a context which is a priori only metric. In this context, the metric jets play the part of the dierential maps. The proofs of the assertions of this section can be found in the rst chapter of [1], in [2] and in [6] ; except for the proof of the fact that the metric jet I is a good jet (see examples 2.1 below) that can be found at the end of this paper. In this section, I recall the concepts which are useful for the understanding of the next section 3. 2.1 The category Jet 0 0 Let (M, a) and (M , a ) be two pointed metric spaces (the chosen points being always assumed to be non isolated). A metric jet (in short jet ) ϕ : (M, a) −→ (M 0 , a0 ) is an equivalence class (of maps f : M −→ M 0 which are LLa * and verify f (a) = a0 ) for the equivalence relation (of tangency at a) : f ≺a g if f (a) = g(a) d(f (x),g(x)) = 0. Indeed, when M and M 0 are n.v.s.*, we come accross and limx→a d(x,a) the usual notion of tangency at a point : more precisely, if f is Diffa *, we have f ≺a Afa where Afa (x) = f (a)+dfa (x − a) is the continuous ane map which is tangent to f at a. 0 0 For such a jet ϕ : (M, a) −→ (M , a ), we are interested in its lipschitzian ratio ρ(ϕ) = inf{k > 0 | ∃f ∈ ϕ, f k -LLa }. This ratio veries the inequality 0 0 ρ(ϕ .ϕ) ≤ ρ(ϕ )ρ(ϕ), the composition of jets being dened just below. The local lipschitzianity is a sucient condition for composing the jets ; indeed, f if M g / /M f0 0 g0 0 / 00 0 00 / M , where M, M , M are metric spaces respecti- 00 0 0 0 vely pointed by a, a , a , the quoted maps verifying f (a) = g(a) = a , f (a ) = g 0 (a0 ) = a00 , and f, g (resp. f 0 , g 0 ) being LLa (resp. LLa0 ), then, we have the implication : f  ≺a g and f 0  ≺a0 g 0 =⇒ f 0 .f ≺a g 0 .g . So, the jets are the morphisms of a 0 category Jet whose objects are the pointed metric spaces (ϕ .ϕ is the jet containing 0 0 0 f .f where f ∈ ϕ and f ∈ ϕ ). This category Jet is cartesian and enriched in Met*, a well-chosen category of metric spaces (whose morphisms are LSL* maps). Thus, 0 0 we can speak of the distance d(ϕ, ψ) when ϕ, ψ ∈ Jet((M, a), (M , a )) as being the quasi-distance d(f, g)* where f ∈ ϕ and g ∈ ψ . 2 37 0 0 0 More precisely, if we denote LL((M, a), (M , a )) the set of maps f : M −→ M 0 which are LLa and verify f (a) = a (providing Hom for a cartesian category deno0 0 0 0 ted LL), and q the canonical surjection LL((M, a), (M , a )) −→ Jet((M, a), (M , a )) = LL((M, a), (M 0 , a0 ))/ ≺a , then the quasi-distance* on LL((M, a), (M 0 , a0 )) factorizes through the quotient giving a distance, dened by d(q(f ), q(g)) = d(f, g). 0 0 We notice that, for all ϕ ∈ Jet((M, a), (M , a )), we have d(ϕ, Oaa0 ) ≤ ρ(ϕ), 0 0 0 where Oaa0 : (M, a) −→ (M , a ) is the jet containing the constant map on a . A jet ϕ is said to be a good jet if the previous inequality is an equality. Examples 2.1 Here, except for example 5 (for which M =] − 1, 1[ and M M = M 0 = R ; and everywhere a = a0 = 0. We set O = O00 . 0 = R), we have 1) O is the jet of all the LL0 maps which are tangent at 0 to the constant map on 0 ; it is a good jet with ρ(O) = 0. 2) V is the jet containing the absolute value v(x) = |x| ; This jet V contains the functions (considering Taylor expansions at order 1) exp |x| − 1, |x|, sin |x|, log(1 + |x|), Arctg|x| . . . etc. It is a good jet since d(V, O) = d(v, 0)* = limr→0 sup{ v(x) | 0 6= |x| ≤ r} = 1 ≤ ρ(V) ≤ 1, this last inequality being |x| due to the fact that v is 1-lipschitzian. S n 3) G is the jet of the Giseh function g(x) = d(x, K∞ ) where K∞ = n∈N 3 K and K is the triadic Cantor set. This jet is a good jet with ρ(G) = 1. 4) F is the jet of the fractal wave function ξ(x) = x sin log |x| if x 6= 0, ξ(0) = 0 ; √ 2. this jet is not a good one since 1 = d(F, O) < ρ(F) = 5) I is the jet of the uncanny function (in french insolite) Ins :] − 1, 1[−→ R : x 7→ x sin log | log |x|| if x 6= 0, Ins(0) = 0 ; we prove (see Proof 1 at the end of this paper) that this jet I is a good one, with ρ(I) = 1. We will meet again these examples farther ; in particular, one can nd the graphs of g and ξ in examples 3.2). We conclude these examples with the jet Ja containing the canonical injection j : V ,→ M , where V is a neighborhood of 3 a in a metric space M ; this jet 38 Ja : (V, a) −→ (M, a) is an isomorphism in Jet, its inverse Ja−1 being the jet of the map s : M −→ V dened by s|V = idV and s(x) = a when x ∈ / V . These −1 jets are good jets with ρ(Ja ) = ρ(Ja ) = 1. 0 0 0 Let (M, a) and (M , a ) be pointed metric spaces, V (resp. V ) a neighborhood of 0 0 0 0 a in M (resp. of a in M ). If ϕ ∈ Jet((V, a), (V , a )), we denote Γ(ϕ) the following composite jet (that we call the stretching of ϕ to M ) : (M, a) Ja−1 / (V, a) ϕ / (V 0 , a0 ) Ja0 / (M 0 , a0 ) 0 0 0 0 This denes an isometry Γ : Jet((V, a), (V , a )) −→ Jet((M, a), (M , a )) which −1 veries ρ(Γ(ϕ)) = ρ(ϕ) (since ρ(Γ(ϕ)) ≤ ρ(Ja0 )ρ(ϕ)ρ(Ja ) = ρ(ϕ) ; same for the inverse inequality), so that Γ(ϕ) is a good jet i ϕ is a good jet. 2.2 Tangent jets Untill now, we only have spoken of jets for maps which are locally lipschitzian at a point. More generally, we can associate a jet to a map which is tangentiable at a point ; the tangentiable maps are natural generalisations of the dierentiable maps. 0 0 Let M, M be two metric spaces, f : M −→ M a map and a ∈ M . We say that f is tangentiable at a (in short T anga ) if there exists an LLa map g : M −→ M 0 such that f  ≺a g . If f is T anga , the set {g : M −→ M 0 | g is LLa and f ≺a g} 0 is a jet (M, a) −→ (M , f (a)) which is denoted Tfa and called the tangent jet of f LLa ⇐⇒ f T anga with f ∈ Tfa (i.e Tfa = q(f )) ; in fact, f LLa =⇒ f T anga =⇒ f LSLa =⇒ f Ca0 *. at a. By denition, f We have a composition of the tangent jets for composable tangentiable maps : T(g.f )a =Tgf (a) .Tfa if f is T anga and g is T angf (a) . Examples 2.2 1) f Diffa * =⇒ f T anga , where Tfa is the jet containing the continuous ane map Afa tangent to f at a. 2) All the examples 2.1 are lipschitzian but not Diff0 : we have v ∈ V =Tv0 , g ∈ G =Tg0 , ξ ∈ F =Tξ0 , Ins ∈ I =TIns0 . We also have j ∈ Ja =Tja . . . etc. 1 if x 6= 0, f1 (0) = 0, is not T ang0 , although obviously LSL0 . 3) f1 (x) = x sin x 1 2 4) f2 (x) = x sin 2 if x 6= 0, f2 (0) = 0, is T ang0 (since it is Diff0 ) but not x 1 LL0 (since limk→+∞ f20 ( √2kπ ) = −∞) ; so that the tangent jet T(f2 )0 exists, but f2 ∈ / T(f2 )0 . 4 39 2.3 Inside n.v.s.* Untill now we have contented ourselves with a purely metric context ; from now on, we will consider all the previous new notions in the n.v.s. frame (which will 0 provide new concepts for such a classical frame). We denote E , E . . . such n.v.s.. LL((E, a), (E 0 , 0)), the set Jet((E, a), (E 0 , 0)) is a 0 vector space (since the vector space (E , 0) is also a vector space internally in Jet : 0 see examples 2.3 below) ; and the canonical surjection q : LL((E, a), (E , 0)) −→ Jet((E, a), (E 0 , 0)) is a linear map. In fact, Jet((E, a), (E 0 , 0)) is a n.v.s., its distance 0 deriving from a norm kϕk = d(ϕ, Oa0 ). Thus ϕ : (E, a) −→ (E , 0) is a good jet i ρ(ϕ) = kϕk. 0 Notably, we nd the following good jets : if l : E −→ E is a continuous 0 linear map, then l is T ang0 with l ∈Tl0 : (E, 0) −→ (E , 0), and the restriction q L(E, E 0 ) ,→ LL((E, 0), (E 0 , 0)) −→ Jet((E, 0), (E 0 , 0)) : l 7→ Tl0 is a linear 0 0 isometric embedding (L(E, E ) being the set of all continuous linear maps E −→ E , kl(xk ). So we have klkop = kq(l)k = equipped with the operator norm klkop = supx6=0 kxk kTl0 k = d(Tl0 , O) ≤ ρ(Tl0 ) ≤ klkop , the last inequality being due to the well-known fact that l is klkop -lipschitzian ; this implies that Tl0 is a good jet. First, we notice that, like Examples 2.3 1) If E is a n.v.s., we denote σ : E×E −→ E : (x, y) 7→ x+y and mλ : E −→ E : x 7→ λx the continuous linear operations of E ; Then, Tσ(0,0) : (E, 0)2 −→ (E, 0) and Tmλ : (E, 0) −→ (E, 0) are good jets, respectively denoted + and µλ . The data of these two jets confers on (E, 0) a structure of vector space, internally in Jet. 2) The translation θab : E −→ E : x 7→ x + b − a provides a jet T(θab )a : (E, a) −→ (E, b) denoted γab which veries ρ(γab ) ≤ 1 and is invertible in Jet with −1 γab = γba . If ϕ ∈ Jet((E, a), (E 0 , a0 )), we denote Ω(ϕ) the following composite jet (that we call the translate of ϕ in 0) : γ0a (E, 0) / (E, a) ϕ / (E 0 , a0 ) γa0 0 / (E 0 , 0) 0 0 0 This denes an isometry Ω : Jet((E, a), (E , a )) −→ Jet((E, 0), (E , 0)) which veries ρ(Ω(ϕ)) = ρ(ϕ), so that Ω(ϕ) is a good jet i ϕ is a good jet. 2.4 Tangentials 0 0 is T anga , where U and U are open subsets of E and E 0 respectively, a ∈ U , we denote tfa : (E, 0) −→ (E , 0) the following composite jet If a map f : U −→ U 0 5 40 (that we call the tangential of f at a) : (E, 0) γ0a −1 / (E, a) Ja / (U, a) Tfa / (U 0 , f (a)) Jf (a) / (E 0 , f (a)) γf (a)0 / (E 0 , 0) In other words, this jet tfa = Ω(Γ(Tfa )) is a streched translate at 0 of the tangent jet Tfa in Jet. Now, if f is tangentiable at every point of U , it provides a 0 map tf : U −→ Jet((E, 0), (E , 0)) : x 7→ tfx , which is called the tangential of f . 2.5 linear jets I complete this section with the notion of linear jet. 0 A jet ϕ : (E, 0) −→ (E , 0) is said to be linear if the following two diagrams commute in the category Jet : (E, 0)2 ϕ2 / (E 0 , 0)2 + (E, 0) (E, 0) µλ   ϕ / (E 0 , 0) µλ +  ϕ / (E 0 , 0) (E, 0)  ϕ / (E 0 , 0) where the jets + and µλ have been dened in examples 2.3. These two commutative diagrams merely mean that the jet ϕ is linear internally 0 0 in Jet. The set of the linear jets (E, 0) −→ (E , 0) is denoted Λ(E, E ) ; 0 If l : E −→ E is a continuous linear map, its tangent jet Tl0 is a good linear 2 jet (just apply the composition of the tangent jets to the equalities l.σ = σ.l 0 and l.mλ = mλ .l ) ; ex : + and µλ are good linear jets, so that the set Λ(E, E ) is 0 a sub-n.v.s. of Jet((E, 0), (E , 0)) ; more precisely, the linear isometric embedding 0 0 L(E, E ) ,→ Jet((E, 0), (E , 0)) : l 7→ Tl0 factorizes through Λ(E, E 0 ). Examples 2.4 1) As previously said, the continuous linear maps give rise to linear jets ; but, as the following example shows it, a linear jet is not necessarily the jet of a continuous linear map (however, we will see in theorem 3.11 that it can be true in a specic context). 2) The tangent jet I =TIns0 (dened in examples 2.2) is not linear (]−1, 1[ being not a vector space) ; but its streching jet Γ(I) to R : (R, 0) J0−1 / (]−1, 1[, 0) TIns0 / (R, 0) is a linear jet. In fact, Γ(I) =TIns0 where Ins is the extent of Ins to R (giving c the value 0 on ] − 1, 1[ ) ; this extent keeps all the local properties of Ins at 0. 6 41 Let us stand still for a while on this uncanny function Ins, just to have a better understanding of the notion of linear jet. The commutativity of the two square diagrams expressing the linearity of the jet Γ(I) simply means that Ins.σ  ≺(0,0) σ.Ins2 and Ins.mλ ≺0 mλ .Ins ; i.e that the uncanny function Ins veries : Ins(x + y) − Ins(x) − Ins(y) =0 (x,y)→(0,0) k(x, y)k lim Ins(λx) − λIns(x) =0 x→0 x lim This could be expressed saying that Ins is linear at the limit at 0 ; let us have a look on the graph of Ins, just to get a good idea of this limit linearity. If, at rst sight, it may seem rather simple, the appearances are however misleading : considering more and more powerful zooms on 0, we notice that, getting closer and closer to 0, the slope is constantly changing (which only means that Ins is not Diff0 ) ; the most important thing being that more we get closer to 0, more the function is rectilinear. And, indeed, more and more rectilinear can be expressed saying linear at the limit ! 2.6 Tangentially linear maps The notion of linear jet gives us the occasion of dening a new generalization of dierentiable maps (of course still in the n.v.s. context). 0 0 If U and U are open subsets of E and E respectively, and if a ∈ U , a map f : U −→ U 0 is said to be tangentially linear at a (in short T La ) if f is T anga and 0 if its tangential at a tfa : (E, 0) −→ (E , 0) is linear (we could say that the tangent jet Tfa is ane). Now, if f is tangentially linear at every point of U , its tangential tf : U −→ Jet((E, 0), (E 0 , 0)) : x 7→ tfx factorizes through Λ(E, E 0 ) ; we denote λf the res0 triction of tf to U −→ Λ(E, E ). 7 42 Examples 2.5 =⇒ f T La with dfa ∈ tfa =T(dfa )0 . For example, f2 (x) = x2 sin x12 if x 6= 0, f2 (0) = 0, is T L0 with t(f2 )0 =T(f2 )0 =T(df2 )0 = 0. 2) The uncanny function Ins :] − 1, 1[−→ R is T L0 , but not Diff0 . 1) f Diffa We end this section with the following local inversion theorem : f : U −→ U 0 be a CT L* map, where U and U 0 are open subsets 0 of E and E (supposed to be Banach spaces) respectively, and a ∈ U . We assume 0 that there exists an invertible germ G : E −→ E in GCTL* such that G ⊂ λfa . 0 Then, there exists an open neigborhood V of a in U such that f (V ) is open in E and that the restriction of f to V −→ f (V ) is invertible in CTL*. Theorem 2.6 Let Here, the invertible germ G verifying G ⊂ λfa plays the part of the invertible 1 dierential dfa (f being then supposed to be of class C ) of the classical local inversion theorem. 3 Representable metric jets I now come to the heart of my subject, still in the simplifying n.v.s. framework (even though it is possible to work in more general specic metric spaces that we call Σ-contracting spaces*). The proofs of the assertions of this section can be found in the second chapter of [1] and in [3] ; except for the proofs of the theorem 3.11 and of the fact that the bifractal wave function is not neofractal at 0 (see examples 3.14) that can be found at the end of this paper. All the basic concepts used here have been recalled in the previous section 2. 3.1 Valued monoid A monoid Σ (whose law is denoted multiplicatively) is called a valued monoid if it is equipped with a specic element 0 and with a homomorphism v : Σ −→ R+ verifying the two conditions : v(t) = 0 ⇐⇒ t = 0 and ∃t ∈ Σ (0 < v(t) < 1). 0 A map σ : Σ −→ Σ is said to be a morphism of valued monoids if it veries σ(0) = 0 and v(t) = v 0 (σ(t)) for all t ∈ Σ. 8 43 Examples 3.1 We will consider the two following examples of morphisms of valued monoids : N0k ,→ R+ ,→ R (with 0 < k < 1), where N0k = {k n | n ∈ N} ∪ {0} is valued by idN0k ; R+ valued by idR+ , and R valued by v(t) = |t|. If E is a n.v.s. (and a ∈ E ), we denote it Ea if we consider it as being centred in a : this Ea is a R-vector space with 0a = a, x +a y = a + ((x − a) + (y − a)) and λ.a x = a + λ(x − a) ; it is even a n.v.s., setting kxka = kx − ak. Of course, E0 = E with its given norm. Now, every valued monoid Σ provides Ea with a new canonical external ope- = a + v(t)(x − a) ; which, besides the usual properties of external operations, veries 0 ?a x = a for all x ∈ Ea and t ?a a = a for all t ∈ Σ ; and is compatible with the norm on Ea i.e veries, for all t ∈ Σ and x ∈ Ea , kt ?a xka = v(t)kxka , which implies limn→+∞ tn ?a x = a when t ∈ Σ veries 0 < v(t) < 1. Besides, for every morphism of valued monoids σ : Σ −→ Σ0 , we have t ?a x = σ(t) ?0a x for all t ∈ Σ and x ∈ Ea . ration by setting t ?a x 3.2 Homogeneous maps In all that follows, Σ is a valued monoid. −→ Ea0 0 is said to be Σ-homogeneous if it veries h(t ?a x) = t ?0a0 h(x) for all t ∈ Σ and x ∈ Ea ; such an homogeneous map veries h(a) = h(0 ?a a) = 0 ?0a0 h(a) = a0 . Thanks to the morphisms of valued monoids v Σ −→ R+ ,→ R, we have the implications : R-homogeneous =⇒ R+ -homogeneous =⇒ Σ-homogeneous for all Σ. 0 If we consider n.v.s. centred in 0, then h : E −→ E is R+ -homogeneous if it veries h(tx) = th(x) for all t ∈ R+ and x ∈ E , i.e h is positively-1-homogeneous ; 0 0 and h : E −→ E is Nk -homogeneous if it veries the fractal property h(kx) = kh(x) 0 for all x ∈ E ; it is why we say k -fractal instead of Nk -homogeneous. Why fractal ? Because of the equivalence : (x, y) ∈ Graph(h) i (kx, ky) ∈ Graph(h), meaning that Graph(h) remains identical to itself when we zoom into 0 with a ratio k (this process being iterated for an innity of times). We can have an approximative idea of a fractal function h : R −→ R, by considering 0 as a point at the innity (i.e at the unreachable horizon point), the graph of h, being then seen A map h : Ea in perspective, innitely decreasing towards this horizon point, and still remaining itself, but thinner and thinner. 9 44 Examples 3.2 1) Linear =⇒ R+ -homogeneous =⇒ Σ-homogeneous, for all Σ. 2) The function v(x) = |x| is well-known to be R+ -homogeneous. 3) The Giseh function g(x) = d(x, K∞ ) where K∞ = 1 triadic Cantor set, is -fractal. 3 3) The fractal wave function e−2π -fractal. Proposition 3.3 If 0 < v(t0 ) < 1. S n n∈N 3 K and K is the ξ(x) = x sin log |x| if x 6= 0, ξ(0) = 0, is h : E −→ E 0 is Σ-homogeneous, then h is v(t0 )-fractal, where Σ-uniqueness property) If h1 , h2 : Ea −→ Ea0 0 are Σ-homogeneous ; then : h1  ≺a h2 =⇒ h1 = h2 . Proposition 3.4 ( h : Ea −→ Ea0 0 is Σ-homogeneous, then : h LLa ⇐⇒ h lipschitzian ⇐⇒ ∃g LLa , g ≺a h ⇐⇒ h T anga ; and then ρ(Tha ) = inf{k > 0|h k -lipschitzian} = supx6=y kh(x)−h(y)k . kx−yk Proposition 3.5 If h : Ea −→ Ea0 0 is Σ-homogeneous, then h is ρ(Tha )-lipschitzian, i.e this lipschitzian ratio ρ(Tha ) is reached in h. Proposition 3.6 If 10 45 Σ-Lhomogeneous will mean lipschitzian Σ-homogeneous. Let us denote Σ-Lhom(Ea , Ea0 0 ) = {h : Ea −→ Ea0 0 |h Σ-Lhomogeneous, h(a) = a0 } ; it is a 0 0 0 subset of LL((E, a), (E , a )). The set Σ-Lhom(Ea , E0 ) is a sub-vector space of LL((E, a), (E 0 , 0)). We denote also Σ-Lhom(E, E 0 ) the vector space Σ-Lhom(E0 , E00 ) ; 0 0 of course, L(E, E ) is itself a sub-vector space of Σ-Lhom(E, E ) for all Σ. Thanks to the Σ-uniqueness property, the restriction of the canonical surjection q to Σ-Lhom(Ea , Ea0 0 ) −→ Jet((E, a), (E 0 , a0 )) is injective, which allows to dene a 0 0 0 distance d(h, h ) = d(Tha ,Tha ) on Σ-Lhom(Ea , Ea0 ). We recall that this distance 0 0 0 on Σ-Lhom(Ea , Ea0 ) was dened at rst as a quasi-distance on LL((E, a), (E , a )). Proposition 3.7 The above distance on supx6=a kh(x)−h0 (x)k kx−ak Σ-Lhom(Ea , Ea0 0 ) can be written d(h, h0 ) = . (x)k and referring to the glossary, we Proof : Denoting δ(h, h0 ) = supx6=a kh(x)−h kx−ak 0 r 0 0 show that, for all r > 0, we have d (h, h ) = δ(h, h ). Indeed, we have immediately r 0 0 d (h, h ) ≤ δ(h, h ). Now, if x ∈ Ea , t ∈ Σ (with 0 < v(t) < 1) and n ∈ N verify ktn ?a xka ≤ r (recalling that limn→+∞ tn ?a x = a), we use the Σ-homogeneousness n 0 (tn ? x)k kh(x)−h0 (x)k a of h to obtain : = kh(t ?ktanx)−h ≤ dr (h, h0 ), which implies δ(h, h0 ) ≤ kx−ak ?a x−ak dr (h, h0 ). Σ-Lhom(E, E 0 ) is a n.v.s. where khk = supx6=0 kh(x)k . This norm kxk 0 0 veries kh(x)k ≤ khk kxk and kh .hk ≤ kh k khk for all x ∈ E , if h ∈ Σ-Lhom(E, E 0 ) and h0 ∈ Σ-Lhom(E 0 , E 00 ). It goes without saying that L(E, E 0 ), equipped with its operator norm, is a sub-n.v.s. of Σ-Lhom(E, E 0 ). Proposition 3.8 q Proof : We just have to use again the isometric embedding Σ-Lhom(E, E 0 ) −→ Jet((E, 0), (E 0 , 0)) : h 7→ Th0 , which here is also linear, to dene the norm khk = kTh0 k = d(Th0 , O) = d(h, 0) on Σ-Lhom(E, E 0 ). 3.3 representable jets The lipschitzian homogeneous maps will represent some metric jets, said to be representable ; having in mind the example of the continuous ane map Afa : E −→ E 0 tangent at a ∈ U to f : U −→ U 0 (supposed to be dierentiable at a ∈ U ; U and U 0 being open subsets of E and E 0 respectively) which plays the 0 starring role in its tangent jet Tfa ∈ Jet((U, a), (U , f (a)) (or, better said, in its streching jet Γ(Tfa ) to E ) : actually, Afa is the unique continuous ane map of the jet Γ(Tfa ) ; thus, in a way, we could say that Afa represents the jet Γ(Tfa ). 11 46 Σ-Lhom(E, E 0 ) −→ Σ-Lhom(Ea , Ea0 0 ) : h 7→ ha , a 0 where h (x) = a + h(x − a), is a translate of h in a, is a bijective isometry. Proposition 3.9 The map : ϕ : (E, a) −→ (E 0 , a0 ) is said to be Σ-representable if there exists h ∈ Σ-Lhom(Ea , Ea0 0 ) such that h ∈ ϕ (i.e, h being T anga , Tha = ϕ) ; which 0 a is equivalent to say that there exists h ∈ Σ-Lhom(E, E ) such that h ∈ ϕ. Such an element of ϕ is unique (thanks to the Σ-uniqueness property) and is called the Σ-representative element of ϕ : it plays a central role in ϕ since, on the one hand ρ(ϕ) is reached in it (see prop. 3.6 ; besides a Σ-representable jet ϕ : (E, 0) −→ (E 0 , 0) is good i ρ(ϕ) = khk) and, on the other hand it gives the direction of ϕ, A jet since it veries the following property : ϕ : (E, a) −→ (E 0 , a0 ) is a Σ-representable jet and if h is its Σ-representative element, then, for all f ∈ ϕ and all x ∈ E , we have : (a) ). h(x) = lim06=v(t)→0 t−1 ?0a0 f (t ?a x) = lim06=v(t)→0 (f (a) + f (a+v(t)(x−a))−f v(t) Proposition 3.10 If ϕ ∈ Jet((E, 0), (E 0 , 0)). Then we have the equivalence : ϕ is the jet of a continuous linear map ⇐⇒ ϕ is a Σ-representable linear jet for a Σ. Hence, the equivalence : ϕ is the jet of a continuous linear map ⇐⇒ ϕ is a linear jet, when ϕ is Σ-representable for a Σ. Theorem 3.11 Let Proof : See the Proof 2 at the end of this paper. 3.4 Contactable maps We now come to the nal generalization of dierentiable maps : the T anga maps f which are contactable at a, their contact at a being a well-chosen map κfa in the tangential tfa (this contact is then the analogous of the dierential dfa ). 0 0 In all that follows, we consider a map f : U −→ U , where U and U are 0 open subsets respectively of E and E , and a ∈ U . Such an f is said to be Σ-contactable at a (in short Σ-Conta ), if f is T anga and if the streching jet Γ(Tfa ) : (E, a) −→ (E 0 , f (a)) is a Σ-representable jet ; which is equivalent to 0 a say that there exists h ∈ Σ-Lhom(E, E ) such that h  ≺a f . Still thanks to the a a Σ-uniqueness property, these h and h are unique, h being the Σ-representative element of Γ(Tfa ), while h, denoted κfa , is called the Σ-contact of f at a ; this κfa is the Σ-representative element of the tangential of f at a, tfa = Ω(Γ(Tfa )). Referring f (a+v(t)x)−f (a) ) for to prop. 3.10, this Σ-contact can be written κfa (x) = lim06=v(t)→0 ( v(t) all x ∈ E . We still have a composition of Σ-contacts for composable Σ-contactable maps : κ(g.f )a = κgf (a) .κfa if f is Σ-Conta and g is Σ-Contf (a) . 12 47 We have the implication : f Diffa =⇒ f Σ-Conta for all Σ with κfa = dfa (the inverse implication being true i the Σ-contact κfa is linear). In fact, applying thm 3.11 to the tangent jet T(κfa )0 , we have the following result : f Diffa ⇐⇒ f T La and f Σ-Conta for a Σ. Hence the equivalence : f Diffa ⇐⇒ f T La , when f is Σ-Conta for a Σ. Theorem 3.12 We have the equivalence : We denote κ+ fa the R+ -contact at a of a map f which is R+ -contactable at a (this R+ -contact is a R+ -Lhomogeneous map). We say that a map f is k -neofractal at a (in short k -neofracta ) if it is 0 Nk -Conta ; its N0k -contact at a (denoted κk fa ) is a k -Lfractal map (i.e a N0k -Lhomogeneous map) ; we denote k -Lfract(E, E 0 ) the n.v.s. N0k -Lhom(E, E 0 ), 0 neofracta (resp. Lfractal and Lfract(E, E )) meaning that such a k ∈]0, 1[ exists. Remarks 3.13 1) Of course, by denition of the contactibility, we have the implication : h : E0 −→ E00 Σ-Lhomogeneous =⇒ h Σ-Cont0 , with κh0 = h. In particular, h k -Lfractal =⇒ h k -neofract0 , with κk h0 = h. 2) Referring to prop. 3.3, we have the implication : f Σ-Conta =⇒ f neofracta . 3) Referring to previous implications, we have the particular implications : f Diffa =⇒ f R+ -Conta =⇒ f neofracta Examples 3.14 1) E being a given n.v.s., every norm on E (which is equivalent to the given norm on E ) is R+ -Cont0 with κ+ n0 = n, since n is R+ -Lhomogeneous ; it is the case for the function v(x) = |x|. S n 2) The Giseh function g(x) = d(x, K∞ ), where K∞ = n∈N 3 K and K is the 1 1 triadic Cantor set, is -neofract0 (even -Lfractal : see example 3.2) with κ 1 g0 = g . 3 3 3 3) The fractal wave function ξ(x) = x sin log |x| if x 6= 0, ξ(0) = 0, is e−2π -neofract0 (even e−2π -Lfractal : see examples 3.2) with κe−2π ξ0 = ξ . However, this function is not R+ -Cont0 , since κe−2π ξ0 = ξ is not R+ -Lhomogeneous. 4) Let us consider the following bifractal wave function dened by ζ(x) = x sin 2π log |x| if x < 0, ζ(x) = x sin 2π log |x| if x > 0, ζ(0) = 0, where a b a, b are > 0 real numbers verifying ab 6∈ Q. We will prove (in Proof 3 at the end of this paper) that this function is T ang0 , but not neofract0 (although for r > 0, ζr (x) = x sin 2π log |x| if x 6= 0, ζr (0) = 0, is e−r -Lfractal !) ; thus, referring to r remarks 3.13, this bifractal wave function is Σ-Cont0 for none Σ. 13 48 5) Let us at last notice that our uncanny function Ins(x) = x sin log | log |x|| if x 6= 0, Ins(0) = 0, is also not Σ-Cont0 for any Σ, since it is T L0 (see examples 2.4 and examples 2.5) and not Diff0 : it thus remains to use thm. 3.12 ! Finaly, the fractal waves allows us to establish the following remarkable result (remarkable, since it deals with jets at order 1 !)) : Theorem 3.15 Jet((R, 0), (R, 0)) is a n.v.s of innite dimension. Proof : Indeed, by a constructing procedure which is analogous to the one −2π of our e - fractal wave ξ(x) = x sin log |x| if x 6= 0, ξ(0) = 0, we associate a −T e -fractal wave f˜(x) = xf (log |x|), f˜(0) = 0, to every function f : R −→ R which is lipschitzian, T periodic (T > 0) and for which there exists a right derivative at each point. Denoting Per(T ) the vector space of these functions f , −T we have an evident embedding Per(T ) −→ e -Lfract(R, R) : f 7→ f˜. We just have now to compose −T e -Lfract(R, R) −→ this embedding with our well-known embedding q : Jet((R, 0), (R, 0)) : h 7→ Th0 to obtain an embedding Per(T ) −→ Jet((R, 0), (R, 0)) ; it remains then to use the fact that Per(T ) is of innite dimension. 4 Contactibility with some classical Theorems Skimming through the metric Differential Calculus has highlighted many gene- ralizations of the specic properties of the classical dierentials. Actually, for contactable maps, we can add to these generalizations a mean value theorem ; and theorems about extrema which, unlike the classical ones, need hypothesis only at order 1 ! In what follows, U and U 0 0 are open subsets of n.v.s. E and E respectively. f : U −→ U 0 be a continuous map, a, b ∈ U such that [a, b] ⊂ U , F a nite subset of ]a, b[. We assume that, for all x ∈ ]a, b[−F , the map f is Σ-Contx and satises kκfx k ≤ k (where k ≥ 0 ; the previous norm has been dened in prop.3.8). Then we have : kf (b) − f (a)k ≤ kkb − ak. Theorem 4.1 Let f : U −→ R be Σ-Conta (with a ∈ U ), admitting a local minimum at a. Then κfa admits a global minimum at 0. Theorem 4.2 Let 14 49 Remark 4.3 This gives back the well-known result of the dierentiable case :  f admits a local minimum at a =⇒ a is a critical point of f (i.e dfa = 0). Indeed, if f is Diffa , then f is Conta with κfa = dfa , so that the Σ-contact κfa is a continuous linear function E −→ R admitting a global minimum at 0 : it forces this κfa to be the null function. f : U −→ R be R+ -Conta (with a ∈ U ; E being here of nite dimension), such that κ+ fa > 0 (i.e verifying κ+ fa (x) > 0 for every x ∈ E − {a}). Then, f admits a strict local minimum at a. Theorem 4.4 Let Remark 4.5 This theorem has not its equivalent, at order 1, in classical Dieren- tial Calculus, since a linear function cannot have a strict minimum. It is rather inspired by theorems giving sucient conditions, at order 2, for the existence of extrema. PROOFS Proof 1 : We prove here that the jet I of the uncanny function Ins, dened in examples 2.1 is a good jet, with ρ(I) = 1. 1 First, we notice that Ins is 2-LL0 (since Ins is odd and, on ]0, [, it veries e |Ins0 (x)| ≤ 1+ | log1 x| ≤ 2. Thus ρ(I) ≤ 2 ; and even ρ(I) ≤ 1+ α1 for all α > 0, since Ins is in fact (1 + α1 )-lipschitzian on ] − e−α , e−α [). Finally, we have ρ(I) ≤ 1, and thus d(I, O) ≤ ρ(I) ≤ 1. It remains to prove that 1 ≤ d(I, O) i.e that 1 ≤ d(Ins, 0) (since I =TIns0 and O =T00 ) ; for this, we use the denition of the quasi-distance r recalled in the below glossary. Well, by denition, d(Ins, 0) = limx→0 d (Ins, 0), |Ins(x)| r where d (Ins, 0) = sup{ | 0 < |x| ≤ r}. We notice that, for all r > 0, there |x| π +2kπ 2 −e k) ≤ r, so that | Ins(x exists k verifying xk = e | = | sin log | log xk || = 1 which xk r implies d (Ins, 0) ≥ 1 for all r > 0. It remains then to do r → 0. Proof 2 : We prove here the theorem 3.11. h be the Σ-representative element of a Σ-representable jet ϕ : (E, 0) −→ (E 0 , 0) ; thus ϕ =Th0 with h ∈ Lhom(E, E 0 ). Let us also assume that 2 this jet is T L0 ; we can then write T(h.σ)(0,0) =Th0 .Tσ(0,0) = ϕ.+ = +.ϕ = 2 2 Tσ(0,0) .Th0 =T(σ.h )(0,0) , i.e h.σ  ≺(0,0) σ.h2 . Now, by the Σ-uniqueness property, 2 we deduce the equality h.σ = σ.h , which gives the linearity of h (since h is continuous) ; that is the end of this proof, since h ∈ ϕ (the inverse implication being Let evident). 15 50 Proof 3 : We prove here that the bifractal wave function dened in examples 3.14, is T ang0 but not neofract0 . 2π Let us consider rst the function ζr (x) = x sin log |x| if x 6= 0, ζr (0) = 0 r 2π ∗ 0 (where r > 0) ; it is derivable on R with |ζr (x)| ≤ kr = 1 + ; thus our bifractal r wave function ζ is sup(ka , kb )-lipschitzian which implies that it is T ang0 . We prove now that ζ cannot be neofract0 . Indeed, if there exists k ∈]0, 1[ for which ζ is k -neofract0 , then there exists a k -Lfractal function g : R −→ R verifying g  ≺0 ζ . Thus, for all x ∈ R∗ , we have (since limn→∞ k n x = 0) : |g(kn x)−ζ(kn x)| limn→∞ = 0. Using the fact that g is k -fractal, we can write g(k n x) = |kn x| n x) n x) k n g(x) for all n ∈ N, so that limn→∞ | g(x) − ζ(k | = 0 which gives limn→∞ ζ(k = x kn x kn x g(x) ζ(kn x) 2π log k log k 2π . Now, for x < 0, = sin(n a + a log |x|) ; so, if we set α = a , x kn x log |x| and xn = sin(2πnα + γ), we have obtained that the sequence (xn ) γ = 2π a g(x) g(x) converges towards . So the cluster set of the sequence (xn ) is reduced to and x x thus cannot be equal to [−1, 1] ; this implies that α ∈ Q (see the lemma 4.6 below). log k ∈ Q. Then ab = αβ ∈ Q, which In the same way (when x > 0), we show that β = b contredicts the hypothesis made on the denition of ζ ! Thus, such a k cannot exist. α ∈ Qc and γ ∈ R. We recall the following results : 2πinα 1) The set {e | n ∈ Z} is dense in S 1 = {z ∈ C | |z| = 1} ; and even, the set 2πinα {e | n ∈ N} is still dense in S 1 . iγ 2) Using the bijective isometry ϕ : C −→ C : z 7→ e z , we obtain that the set i(2πnα+γ) 1 {e | n ∈ N} is also dense in S . 3) Setting xn = sin(2πnα + γ) for all n ∈ N, we deduce, not only that the set {xn | n ∈ N} is dense in [−1, 1], but, even better, that the cluster set of the sequence (xn )n∈N is equal to [−1, 1]. Lemma 4.6 Let GLOSSARY n.v.s. : R-normed vector space, usually denoted E . p.m.s. : pointed metric space, usually denoted (M, a) (where a is not isolated). Ca0 : continuous at a (C 0 : continuous). LLa : locally lipschitzian at a. LSLa : locally semi-lipschitzian at a (knowing that f : M −→ M 0 is SLa if ∃k > 0 ∀x ∈ M d(f (x), f (a)) ≤ kd(x, a)). We have the implications : f LLa =⇒ f LSLa =⇒ f Ca0 . 16 51 LL((M, a), (M 0 , a0 )) : the set of maps f : M −→ M 0 which are LLa and which 0 verify f (a) = a . These sets are the Hom of a cartesian category LL. Jet((M, a), (M 0 , a0 )) = LL((M, a), (M 0 , a0 ))/ ≺a ; these sets are the Hom of the cartesian category Jet (whose objects are the p.m.s. and the morphims the metric jets (or jets)). This category Jet is a quotient of the category LL by the relation of tangency. The canonical surjection q : LL −→ Jet is a cartesian functor. Met : the category whose objects are the metric spaces and morphisms the LSL maps. Jet is enriched in Met : for f, g ∈ LL((M, a), (M 0 , a0 )), there exists a k > 0 and a neighborhood V of a on which f and g are k -lipschitzian ; we d(f (x),g(x)) r r | a 6= x ∈ rst dene d(f, g) = limr→0 d (f, g), where d (f, g) = sup{ d(x,a) 0 B (a, r) ∩ V }. It is not a distance since we only have d(f, g) = 0 ⇐⇒ f ≺a g (it is only a quasi-distance) ; this leads us to dene a true distance on the quotient Jet((M, a), (M 0 , a0 )) by setting d(q(f ), q(g)) = d(f, g). T anga : tangentiable at a ; Tfa being the tangent jet at a of f (assumed to be T anga ). If f is LLa , we have f ∈Tfa = q(f ). 0 0 0 In what follows, f : U −→ U , where U and U are open subsets of n.v.s. E and E respectively ; and a ∈ U . Diffa : dierentiable at a ; dfa being the dierential at a of f (assumed to be Diffa ). tfa : tangential at a of f (assumed to be T anga ). CT : continuously tangentiable, i.e tangentiable at every point of U and the map 0 tf : U −→ Jet((E, 0), (E , 0)) : x 7→ tfx , called the tangential of f , is continuous. T La : tangentially linear at a. CT L : continuously tangentially linear i.e tangentially linear at every point of U and the restriction λf of tf to U −→ Λ(E, E 0 ) is continuous. We have the implications : C 1 =⇒ CT L =⇒ CT =⇒ C 0 . L(E, E 0 ) : the set of all continuous linear maps E −→ E 0 . Λ(E, E 0 ) : the set of all linear jets (E, 0) −→ (E 0 , 0). CTL : the category whose objects are the open subsets of n.v.s. and whose morphisms are the CT L maps. GCTL : the category whose objects are the n.v.s., and morphisms E −→ E 0 0 are germs at 0, of maps f : E −→ E verifying f (0) = 0 and for which there exists 0 a neigborhood V of 0 such that f |V : V −→ E is CT L. 17 52 Σ : valued monoid (its valuation being denoted v : Σ −→ R+ ). Σ-Conta : Σ-contactable at a. Σ-contracting space : metric space M , centred in ω , on which Σ externally operates, this operation verifying 0?x = ω for all x ∈ M and t?ω = ω for all t ∈ Σ ; and being compatible with the distance of M , i.e verifying d(t?x, t?y) = v(t)d(x, y) for all (t, x, y) ∈ Σ × M × M . Références [1] E.Burroni and J.Penon, Elements for a metric tangential calculus, http ://fr.arxiv.org/abs/0912.1012 [2] E.Burroni and J.Penon, A metric tangential calculus, TAC 2010 Vol 23 (10) [3] E.Burroni and J.Penon, Representation of metric jets, Cahiers Top.Géo.Di.Cat Vol LI-3 [4] E.Burroni and J.Penon, Calcul diérentiel revisité à la lumière d'une fonction insolite, talk at the SIC in Paris (Oct 2009), http ://people.math.jussieu.fr/ guitart/docpp/ sic24octobre2009paris.pdf [5] E.Burroni and J.Penon, Revisiting Dierential Calculus in the light of an Uncanny function, talk at the CT2010 in Genova (June 2010), http ://ct2010.disi.unige.it/index.php ?page=4 [6] E.Burroni and J.Penon, Revisiting Dierential Calculus in the light of an Uncanny function, JPAA 4572 18