Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary at infinity of the universal cover is endowed with an invariant total cyclic order. Then we study these new objects on surfaces endowed with half-translation structures and on finite metric graphs. The main result of the paper is a theorem of classification of geodesic laminations on a compact surface endowed with a half-translation structure. We also show that every finite metric fat graph, outside four homeomorphism classes, is the support of a geodesic lamination with uncountably many leaves none of which is eventually periodic. 1
In this paper we present our work on shape optimization for soft robotics where the shape is optimized for a given soft robot usage. To obtain a parametric optimization with a reduced number of parameters, we rely on an approach where the designer progressively refines the parameter space and the fitness function until a satisfactory design is obtained. In our approach, we automatically generate FEM simulations of the soft robot and its environment to evaluate a fitness function while checking the consistency of the solution. Finally, we have coupled our framework to an evolutionary optimization algorithm, and demonstrated its use for optimizing the design of a deformable leg of a locomotive robot.
Dans cet article, on construit une compactification équivariante de l'espace P Flat(Σ) des classes d'homothétie de structures de demi-translation sur une surface Σ compacte, connexe, orientable. On définit l'espace P Mix(Σ) des classes d'homothétie de structures mixtes sur Σ, qui sont des structures arborescentes, au sens de Drutu et Sapir, CAT(0), dont les pièces sont des arbres réels ou des complétés de surfaces munies de structures de demi-translation.En munissant Mix(Σ) de la topologie de Gromov équivariante, et en utilisant des techniques de cônes asymptotiques à la Gromov, on montre que P Mix(Σ) est une compactification équivariante de P Flat(Σ), ce qui nous permet de comprendre géométriquement les dégénérescences de structures de demi-translation sur Σ. On compare ensuite cette compactification à celle de Duchin-Leininger-Rafi, qui utilise des courants géodésiques, en passant par les distances de translation des éléments du groupe de revêtement de Σ.Abstract : In this paper, we give an equivariant compactification of the space P Flat(Σ) of homothety classes of half-translation structures on a compact, connected, orientable surface Σ. We introduce the space P Mix(Σ) of homothety classes of mixed structures on Σ, that are CAT(0) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are R-trees and completions of surfaces endowed with half-translation structures.Endowing Mix(Σ) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that P Mix(Σ) is an equivariant compactification of P Flat(Σ), thus allowing us to understand in a geometric way the degenerations of half-translation structures on Σ. We finally compare our compactification to the one of Duchin-Leininger-Rafi, based on geodesic currents on Σ, by the mean of the translation distances of the elements of the covering group of Σ. 1The goal of this paper is to construct and to describe a geometric compactification, natural under the action of the mapping class group, of the space of homothety classes of halftranslation stuctures on a compact surface, endowed with the equivariant Gromov topology. It is part of the wide field of study of deformations of geometric structures on surfaces (see for instance [Gol]). Let Σ be a compact, connected, orientable surface of genus g > 2, without boundary for simplicity (see [Mor1]) for the general case). After the founding fathers Gauss and Riemann who have studied conformal geometry on surfaces, the Teichmüller spaces T (Σ) of isotopy classes of hyperbolic metrics on Σ, have been studied by for instance Fricke, Klein, Fenchel, Nielsen, and the moduli spaces of real projective structures by for instance Goldman and Choi [CG]. The analysis of the space Flat(Σ) of half-translation structures on a Σ is currently blooming, with the works notably of Calta, Eskin, Hubert, Lanneau, Masur, McMullen, Möller, Myrzakhani, Schmidt, Smillie, Veech, Weiss, Yoccoz and Zorich. When these deformations spaces are non compact, it is important and useful to consider the asymptotic behavior ...
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