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Register a new userLa conjecture de Herman
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Mauricio Garay
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In the Nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophatine critical point of a hamiltonian function. I construct a formalism for the UV-cutoff and prove a generalised KAM theorem which solves positively the Herman conjecture.
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In the nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophantine critical point of a hamiltonian function. I show that KAM versal deformation theory solves positively this conjecture.
In this note, we extend the renormalization horseshoe we have recently constructed with N. Goncharuk for analytic diffeomorphisms of the circle to their small two-dimensional perturbations. As one consequence, Herman rings with rotation numbers of bounded type survive on a codimension one set of parameters under small two-dimensional perturbations.
Les travaux presentes dans ce memoire portent sur la dynamique de diffeomorphismes de varietes compactes. Pour letude des proprietes generiques ou pour la construction dexemples, il est souvent utile de savoir perturber un syst`eme. Ceci soul`eve generalement des probl`emes delicats : une modification locale de la dynamique peut engendrer un changement brutal du comportement des orbites. En topologie C^1, nous proposons diverses techniques permettant de perturber tout en contr^olant la dynamique : mise en transversalite, connexion dorbites, perturbation de la dynamique tangente, realisation dextensions... Nous en tirons diverses applications `a la description de la dynamique des diffeomorphismes C^1-generiques. <p> This memoir deals with the dynamics of diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamic may cause a radical change in the behavior of the orbits. For the C^1 topology, we propose various techniques which allow to perturb while controlling the dynamic: setting in transversal position, connection of orbits, perturbation of the tangent dynamics,... We derive various applications to the description of the C^1-generic diffeomorphisms.
This survey article is the written version of a talk given at the Bourbaki seminar in April 2021. We give an introduction to Zagiers conjecture on special values of Dedekind zeta functions, and its relation to $K$-theory of fields and the theory of motives. We survey recent progress on the conjecture and in particular the proof of the $n=4$ case of the conjecture by Goncharov and Rudenko.
The research assessments of countries or institutions should reveal their contribution to the advancement of science. Taking into consideration the correlation that exists between scientific impact and number of citations, research assessments can be based on citation counts. However, it is crucial to perform the counts in the heavy tail of citation distributions. Following this heavy tail approach, all measurements performed in Spanish universities show that their research is inefficient because the proportion of highly cited papers is much lower than in the most scientifically advanced countries. The same conclusion is reached when considering the number of researchers in the Ioannidis, Boyack, and Baas lists of highly cited researchers.
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