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Configurations to Branched Configurations and Beyond Jean-Pierre Magnot

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Configurations to Branched Configurations and Beyond Jean-Pierre Magnot From Configurations to Branched Configurations and Beyond Jean-Pierre Magnot To cite this version: Jean-Pierre Magnot. From Configurations to Branched Configurations and Beyond. Research and Reports on Mathematics, 2017, 1. hal-01834883 HAL Id: hal-01834883 https://hal.archives-ouvertes.fr/hal-01834883 Submitted on 11 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Magnot, Korzh TN, Res Rep Math 2017, 1:1 Research and Reports on Mathematics Research Article a SciTechnol journal in optimal transport, the space of probability measures, and shows From Configurations to how they can also furnish configurations for uncompressible fluids. In these settings, branching effects are well-known and sometimes Branched Configurations and obvious, and we do not need any adaptation of the framework to obtain a full description of them. Therefore, what we call branched Beyond configurations appears as an intermediate (an we hope useful) step Jean-Pierre Magnot* between dynamics of e.g. a N-body problem and e.g. wave dynamics. Dirac configuration spaces on a locally compact manifold Abstract Let us describe step by step a way to build infinite configurations, We propose here a geometric and topological setting for the study as they are built in the mathematical literature. We explain each step of branching effects arising in various fields of research, e.g. in with the configurations already defined in e.g. Albeverio et al. [7] and statistical mechanics and turbulence theory. We describe various Fadell et al. [1], the generalization will be discussed later in this paper. aspects that appear key points to us, and finish with a limit of such A set of I -configurations is a set of objects that are modelizations a construction which stand in the dynamics on probability spaces where it seems that branching effects can be fully studied without of physical quantities. For example, in the settings [1-6], the any adaptation of the framework. physical quantity modeled is the position of one particle. The whole world is modeled as a locally compact manifold N, and the set Keywords of 1-configurations is itself N, or equivalently the set of all Dirac Finite and infinite configuration spaces; Branched dynamics measures on N. Let I be a set of indexes. I can be countable or uncountable. We Introduction define the indexed (or the ordered if I isequipped with its total order) configuration spaces. For this, we need to define a symmetric binary Finite and infinite configuration spaces are rather old topics, see relation relation u on Γ1, that expresses the compatibility of two e.g. [1,2] and the references cited therein, that had many applications physical quantities. We assume also that u has the following property: in various settings in mathematical physics and representation theory. More recently, several papers, including [3-6] showed how ∀(,u v ) ∈Γ (12 ), uUυυ ⇒ u ≠ these topics could be applied in various disciplines: ecology, financial In the settings Albeverio et al. [7] and Finkelshtein et al. [1], two markets, and so on. This large spectrum of applications principally particles cannot have the same position. Then, for x, y N [2], comes from the simplicity of the model: considering a state space N, finite or infinite configurations are finite or countable sets of values in xuy⇔≠ x y ∈ N: This is why we begin with giving a short description of this setting, With these restrictions, we can define the indexed or ordered and describe a differentiable structure that can fit with easy problems configuration spaces: of dynamics. This structure, which can be seen either as a Frölicher nn1 structure or as a diffeological one, is carefully described and the links OΓ={( u11 ,..., unj )∈Γ ( )}such that, ifi ≠ j; uuu between these two frameworks are summarized in the appendix. OOΓ= Γn We also give a result that seems forgotten in the past literature: the n∈* I II infinite configuration space used in e.g. [7] is an infinite dimensional OΓ{( unnI )∈ ∈Γ ( ) such that, ifi≠ j; u1 uu j} manifold. The general configuration spaces are not ordered. Letn Σ be the But the main goal of this paper is to include one dimensional group of bijections on n, and ΣI be the set of bijections on I. We can turbulence effects (in particular period doubling, see e.g. [8]) in the define two actions: dynamics described by finite or infinite configuration spaces. For this, nn ×Γ→OO Γ we change the metric into the Hausdorff metric. This enables to “glue ∑ n together” two configurations into another one and to describe shocks. (σ, (u1,…,un)) →( uσ (1),…, uσ (n)) Therefore, the dynamics on this modified configuration space are and its infinite analog: described by multivalued paths that are particular cases of graphs on ×Γ→II Γ N; which explains the terminology: “branched configuration”. ∑ I OO We finish with the description of the configurations used e.g. (σ, (un)n I) →( uσ (n),) n I ∈ ∈ where ΣI is a subgroup of the group of bijections of I: In the sequel, I is countable with discrete topology, which avoids topological *Corresponding author: Jean-Pierre Magnot, Department of Mathematics, problems on Σ as in more complex examples. Then, we define general University of Angers, Lycee Jeanne dArc, 30 avenue de Grande Bretagne, I F-63000, Clermont-Ferrand, France, Tel: +33 2 41 96 23 23; E-mail: configuration spaces: [email protected] nn Γ=O Γ/ ∑n Received: July 19, 2017 Accepted: October 18, 2017 Published: November 02, 2017 All articles published in Research and Reports on Mathematics are the property of SciTechnol, and is protected by copyright International Publisher of Science, laws. Copyright © 2017, SciTechnol, All Rights Reserved. Technology and Medicine Citation: Magnot JP (2017) From Configurations to Branched Configurations and Beyond. Res Rep Math 1:1 Γ= Γn smooth locally compact manifold. The 1-configurations considered n∈* keep their topological properties, as in the model of elastodymanics (see e.g. Hughes et al. [10]) or in various quantum field theories. Γ=II Γ ∑ O / I Notice also that we do not give compatibility conditions between Configuration spaces on more general settings two 1-configurations: we would like to give the more appropriate conditions in order to fit with the applied models, this is why we leave In the machinery of the last section, the properties of the base this point to more specialized works. manifold N are not used in the definition of the space Γ1. This is why the starting point can be Γ1 instead of N, and we can give it the most Topological 1-configurations general differentiable structure. Let us first consider the most general We follow here, for example, Hughes T et al. [10]. case [9]: 1 n I Definition 2.1: Let M be a smooth compact manifold and N an Proposition 1.1: If Γ is a diffeological space, then Γ , Γ and Γ are arbitrary manifold. We set diffeological spaces Γ=I (,)MN C∞ (,) MN Proof: (Γ1)n (resp. (Γ1)) is a diffeological space according to e Proposition 4.8 (resp. Proposition 4.10), so that, OΓn (resp. OΓI) is One can also only consider embeddings, and set: 1 n 1 n n a diffeological space as a subset of (Γ ) (resp. (Γ ) ). Thus, Γ = O / Γ=I 1 n m (,)M N Emb (,) M N Σn (resp. Γ = O /Σ) has the quotient diffeology by Proposition 4.12, which ends the proof. where dimM<dimN, and Emb is the set of embeddings. The things ∞ I run as in the first case, since Emb(M;N) C (M;N) is an open subset Let us now turn to the cases where Γ has a stronger structure. We ∞ n 1 of C (M;N): already know that Γ is a manifold if Γ is a manifold. ⊂ Proposition 1.2: If Γ1 is a Frölicher space, then Γn (and hence Γ) Examples of topological configurations is a Frölicher space. Links. Let ΓI = Emb(S1;N) Here, we fix the uncompatibility relation as Proof: Adapting the last proof, using Proposition 4.9 instead of n 1 Proposition 4.8, we get that OΓ is a Frolicher space if Γ is a Frolicher γγu ′′⇔ γ(S11 ) γ (S ) ≠∅ space. Let us now build a generating set of functions for the Frolicher n n n Γ k structure on OΓ . Let f: OΓ → be a smooth map. We define the Then, link is the space of n→links of class C , which is a Frechet symmetrization of f: manifold. n 1 f: ( u1 ,..., unn )∈Γ O fu (1 ,..., u )= ∑ fu (σσ(1) ,..., u ( n ) ) Triangulations: Consider the n-simplex n!σ ∈∑n n The set of functions {}f generate the contours on OΓ , and then, ∆=(tt ,..., ) ∈ | t = 1 n nn ∑ i passing to the quotient, generate the contours on Γ Setting a Frölicher structure on indexed configurations is only a If N is a n-dimensional manifold, a (finite) triangulation σ of N straightforward consequence of Proposition 4.10: is such that: 1 I 1 ||σ Proposition 1.3: If Γ is a Frölicher space, then OΓ is a Frölicher space. 1) σ ∈Γ(mn ( ∆ ,N )) But the problem of a Frölicher structure on ΓI a little bit more (2) Let (T, T′) σ2 such that T≠ T′ then Im(T)∩ Im(T′) is a simplex complicated; let us explain why and give step by step the construction or a collection of simplexes of each border Im(∂T) and Im(∂T′): ∈ of the Frölicher structure.
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