Archives Séminaire Analyse

Soit $f$ un endomorphisme du plan projectif complexe de degré $d>1$. Son entropie topologique est $2\log(d)$ (Misiurewicz-Przytycki, Gromov) et $f$ possède une unique mesure d’entropie maximale notée $\mu$. Cette mesure est ergodique et possède deux exposants de Lyapunov qui sont minorés par $(\log d)/2$ (Briend-Duval). La mesure $\mu$ est aussi l’auto-intersection $\mu = T \wedge T$ du courant de Green $T$ de $f$. Les exposants de Lyapunov sont égaux à $(\log d)/2$ si et seulement si $\mu \lll \text{Leb}$ (Ledrappier). C'est équivalent à dire que $T$ est lisse $>0$ sur un ouvert, et c’est aussi équivalent à dire que $f$ est un exemple de Lattès (Berteloot-Loeb, Berteloot-Dupont).

Il est naturel de se demander si l’on peut caractériser de façon similaire la minimalité d’un seul exposant. R. Dujardin a démontré (2012) que l’absolue continuité de $\mu$par rapport à la trace de $T$ implique qu’un exposant est égal à $(\log d)/2$. J’ai ensuite démontré que la réciproque est vraie, donnant une première caractérisation. Les exemples connus d'applications possédant un seul exposant minimal sont donnés par des applications préservant un pinceau de droites avec une dynamique de type Lattès sur le pinceau. À partir d’une relation d'absolue continuité entre $\mu$ et $T$ (en un sens fort), on peut démontrer l’existence d’un tel pinceau avec une dynamique Lattès.

Les démonstrations sont basées sur l'utilisation de formes normales pour la dynamique, ainsi que sur la théorie du pluripotentiel. Pendant l'exposé, je reviendrai sur ces différentes notions.

https://www.math.u-bordeaux.fr/~skupin/conf-pthomas-2024.html

Présentation des membres de l'équipe

In the first part of the talk, we introduce the concept: Controllability of Differential Equations. Then we give some examples in finite (ODE) and infinite dimensional(PDE) contexts. We recall the controllability results of the Transport and Heat equation.

In the second part of the talk, we consider compressible Navier-Stokes equations in one dimension, linearized around a positive constant steady state . It is a Coupled system of Transport (for density) and Heat type (for velocity) equations.  We study the boundary null-controllability  of this linearized system in an interval when a Dirichlet control function is acting either only on the density or only on the velocity component at one end of the interval. In this setup, we state some new control results which we have obtained. We see that these controllability results are optimal/sharp concerning the regularity of initial states (in the velocity case) and time (in the density case). The proof is based on a spectral analysis and on solving a mixed parabolic-hyperbolic moments problem and a parabolic hyperbolic joint Ingham-type inequality.  This is a joint work with Kuntal Bhandari, Rajib Dutta and Jiten Kumbhakar. Finally, the talk ends with some ongoing and future directions of research.

Holomorphic dynamics studies the evolution of complex manifolds under the iteration of holomorphic maps.

While significant progress has been made in understanding the theory of one-dimensional holomorphic dynamics, the transition to higher dimensions still presents difficult challenges since the situation is vastly different from the one-dimensional case.

Even only the study of the dynamics of automorphisms (i.e. holomorphic maps injective and surjective) in two dimensions already poses deep difficulties, and the construction of significant examples is an active area of research.

In this talk, we provide an overview of the dynamics in several complex variables, focusing particularly on the stable dynamics of automorphisms of C^2. We introduce concepts such as Fatou sets, polynomial and transcendental Hénon maps, and limit functions. Finally, we address two recently resolved questions that refer to the current state of my research  (a joint work with A. M. Benini and A. Saracco):

Can limit sets for (non-recurrent) Fatou components be hyperbolic?

Can limit sets be distinct?

The main tool of soliton theory (aka completely integrable systems) is the inverse scattering transform (IST) which relies on solving the Faddeev-Marchenko integral equation. The latter amounts to inverting the I+Hankel operator which historically was done by classical techniques of integral operators and the theory of Hankel operations was not used. In the recent decade however the interest in the soliton community has started shifting from classical initial conditions of integrable PDEs to more general ones (aka none classical initial data) for which the classical IST no longer works. In this talk, on the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation, we show how the classical IST can be extended to serve a broad range of physically interesting initial data. Our approach is essentially based on the theory of Hankel operators.

Given an inner function $\Theta \in H^\infty(\mathbb D)$ and $[g]$ in the quotient algebra $H^\infty/ \Theta H^\infty$,

its quotient norm is 

$\|[g]\|:= \inf \left\{ \|g+\Theta h\|_\infty, h \in H^\infty \right\}$. We show that 

when $g$ is normalized so that $\|[g]\|=1$, the quotient norm of its inverse can be made 

arbitrarily close to $1$ by imposing $|g(z)|\ge 1- \delta$ when $\Theta(z)=0$, with $\delta>0$ small enough, 

(call this property SIP)

 if and only if the function $\Theta$ satisfies the following growth property:

$$

\lim_{t\to 1} \inf\left\{ |\Theta(z)|: z \in \mathbb D, \rho(z, \Theta^{-1} \{0\} ) \ge t \right\} =1,

$$

where $\rho$ is the usual pseudohyperbolic distance in the disc, $\rho(z,w):= \left| \frac{z-w}{1-z\bar w}\right|$.

We prove that an inner function is SIP if and only if for any $\eps>0$, the set $\{ z: 0< |\Theta (z) | < 1-\eps\}$

cannot contain hyperbolic disks of arbitrarily large radius.

Thin Blaschke products provide an example of such functions. Some SIP Blaschke products fail to be interpolating

(and thus aren't thin), while there exist Blaschke products which are interpolating and fail to be SIP. 

We also study the functions which can be divisors of SIP inner functions.

We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on the Hardy space, in terms of the distance to its spectrum $\sigma(T_b)$. We are primarily interested in the case when the symbol $b$ is a Laurent polynomial (\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.

This is a joint work with S. Kupin and A. Vishnyakova.

The Beurling--Selberg extremal approximation problems aim to find optimal unisided bandlimited approximations of a target function of bounded variation. We present an extension of the Beurling--Selberg problems, which we call “of higher-order,” where the approximation residual is constrained to faster decay rates in the asymptotic, ensuring the smoothness of their Fourier transforms. Furthermore, we harness the solution’s properties to bound the extremal singular values of confluent Vandermonde matrices with nodes on the unit circle. As an application to sparse super-resolution, this enables the derivation of a simple minimal resolvable distance, which depends only on the properties of the point-spread function, above which stability of super-resolution can be guaranteed.

Cf. https://plmbox.math.cnrs.fr/f/136ed3186ea241e8b980/