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DynamIC Seminars (Complete List)

Name Title Date Time Room
Pierre Berger (CNRS-LAGA, Université Paris 13, UPSC)Emergence and Para-DynamicsAbstract: Recently we showed that some degenerate bifurcations can occur robustly. Such a phenomena enables ones to prove that some pathological dynamics are not negligible and even typical in the sens of Arnold-Kolmogorov. More precisely, we proved: Theorem: For every $\infty>r\ge 1$, for every $k\ge 0$, for every manifold of dimension $\ge 2$, there exists an open set $\hat U$ of $C^r$-$k$-parameters families of self-mappings, so that for every topologically generic family $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the mapping $f_a$ displays infinitely many sinks. We will introduce the concept of Emergence which quantifies how wild is the dynamics from the statistical viewpoint, and we will conjecture the local typicality of super-polynomial ones in the space of differentiable dynamical systems. For this end, we will develop the theory of Para-Dynamics, by giving a negative answer to the following problem of Arnold (1992): Theorem: For every $\infty>r\ge 1$, for every $k\ge 0$, for every manifold of dimension $\ge 2$, there exists an open set $\hat U$ of $C^r$-$k$-parameters families of self-mappings, so that for every topologically generic family $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the map $f_a$ displays a fast increasing number of periodic points: $$\limsup \frac{\log Card \; Per_n \, f_a}n = \infty$$ We also give a negative answer to questions asked by Smale 1967, Bowen in 1978 and by Arnold in 1989, for manifolds of any dimension $\ge 2$: Theorem: For every $\infty\ge r\ge 1$, for every manifold of dimension $\ge 2$, there exists an open set $U$ of $C^r$-diffeomorphisms, so that a generic $f\in U$ displays a fast growth of the number of periodic points. The proof involves a new object, the $\lambda$-$C^r$-parablender, the Renormalization for hetero-dimensional cycles, the Hirsh-Pugh-Shub theory, the parabolic renormalization for parameter family, and the KAM theory. Tuesday, 17 January 2017 14:00 Huxley 340
Matthieu Astorg (Université d'Orleans)Summability condition and rigidity for finite type mapsAbstract: Finite type maps are a class of analytic maps on complex 1-manifolds introduced by Epstein, that notably include rational maps and entire functions with a finite singular set. Each of those maps possess a natural finite-dimensional moduli space, and one can define a dynamical Teichmüller space parametrizing their quasiconformal conjugacy class. Using the fact that this Teichmüller space immerses into the moduli space, we will generalize rigidity results of Avila, Dominguez, Makienko and Sienra under an assumption of expansion along the critical orbits. Tuesday, 31 January 2017 14:00 Huxley 340
Matteo Ruggiero (Paris 7)Local dynamics of non-invertible selfmaps on complex surfacesAbstract: We consider the local dynamical system induced by a non-invertible selfmap f of C^2 fixing the origin. Given a modification (composition of blow-ups) over the origin, the lift of f on the modified space X defines a meromorphic map F. We say that F is algebraically stable if for every compact curve E in X, its orbit through F does not intersect the indeterminacy set of F. We show that, starting from any modification, we can also blow-up some more and obtain another modification for which the lift F is algebraically stable. The proof relies on the study of the action f_* induced by f on a suitable space of valuations V. In particular we construct a distance on V for which f_* is non-expanding. This allows us to deduce fixed point theorems for f_*. If time allows, I will comment on the recent developments about local dynamics on normal surface singularities. Joint work with William Gignac. Tuesday, 14 February 2017 14:00 Huxley 340
Bassam Fayad (Institut de Mathématiques de Jussieu-Paris Rive Gauche)TBAAbstract: TBA Tuesday, 21 February 2017 14:00 Huxley 340
Ian Melbourne (University of Warwick)TBAAbstract: TBA Tuesday, 28 February 2017 2:00 Huxley 340

DynamIC Workshops and Mini-Courses (Complete List)

Title Date Venue
Mini-Workshop on the Dynamics of Complex NetworksTuesday, 10 January 2017Imperial College London
One Day Dynamics MeetingTuesday, 13 December 2016Imperial College London
Parameter Problems in Analytic DynamicsMonday, 27 June 2016 – Friday, 1 July 2016Imperial College London

Short-term DynamIC Visitors (Complete List)

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