Related papers: Uniform a priori bounds for neutral renormalizatio…

The control of postcritical sets of quadratic polynomials with a neutral fixed point is a main ingredient in the remarkable work of Buff and Ch\'eritat to construct quadratic Julia sets with positive area. Based on the Inou-Shishikura…

We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disc. After normalisation this is a one parameter…

In a companion paper [On semiclassical orthogonal polynomials via polynomial mappings, J. Math. Anal. Appl. (2017)] we proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations. In this work we…

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\Z/N\Z$ introduced by Gowers in his proof of Szemer\'edi's…

We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix…

A while ago MLC (the conjecture that the Mandelbrot set is locally connected) was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at…

We consider the continuum parabolic Anderson model (PAM) and the dynamical $\Phi^4$ equation on the $3$-dimensional cube with boundary conditions. While the Dirichlet solution theories are relatively standard, the case of Neumann/Robin…

We study quasi-periodic eigenvalue problems that arise in the stability analysis of periodic traveling wave solutions to Hamiltonian PDEs. We establish bounds on regions in the complex plane when the eigenvalues may deviate from the…

Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show…

We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical…

Let $p_{\min}$ denote the minimum of a polynomial $p$ over a (general) compact semialgebraic set $S \subseteq \mathbb{R}^n$. A standard way to approximate $p_{\min}$ is via hierarchies built from Positivstellens\"atze, which certify…

Let $X$ be a one dimensional positive recurrent diffusion with initial distribution $\nu$ and invariant probability $\mu$. Suppose that for some $p> 1$, $\exists a\in\R$ such that $\forall x\in\R, \E_x T_a^p<\infty$ and $\E_\nu…

Let $1<p<+\infty$ and let $\Omega\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the…

In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural…

We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite)…

We ask questions generalizing uniform versions of conjectures of Mordell and Lang and combining them with the Morton--Silverman conjecture on preperiodic points. We prove a few results relating different versions of such questions.

In this paper, we first construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue $\tilde{\sigma}_1$ tends to infinity. We then prove that as $g\rightarrow \infty$, a…

Given a reproducing kernel $k$ on a nonempty set $X$, we define the reproductive boundary of $X$ with respect to $k$. Furthermore, we generalize the well known nontangential and horocyclic approach regions of the unit circle to this new…

We introduce the new concepts of pseu\-do numerical range for operator functions and families of sesquilinear forms as well as the pseu\-do block numerical range for $n \times n$ operator matrix functions. While these notions are new even…

We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum…