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The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various…

Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the…

In this article, we develop a functional-analytic framework to establish existence, uniqueness, regularity of disintegration, and statistical properties of equilibrium states for a broad class of dynamical systems, potentially discontinuous…

We study the dynamics of the attractor of the doubling map with an asymmetrical hole by associating to each hole an element of the lexicographic world. A description of the topological entropy function is given. We show that the set of…

In the first part of the paper, we introduce the Hamiltonian $-\Delta-Z/\sqrt{x^2+y^2}$, Z>0, as a selfadjoint operator in $L^2(R^2)$. A general central point interaction combined with the two-dimensional Coulomb-like potential is…

This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order and whose derivative satisfies some growth condition at infinity. This class contains most of the classical families of transcendental…

For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good…

Stable accessibility for partially hyperbolic diffeomorphisms is central to their ergodic theory, and we establish its \(C^1\)-density among 1. all, 2. volume-preserving, 3. symplectic, and 4. contact partially hyperbolic flows. As…

This note is a summary of the recent paper [9]. Here, we study the homogenization of elliptic systems with Dirichlet boundary condition, when both the coefficients and the boundary datum are oscillating. In particular, in the paper [9], we…

In this paper, we first prove an abstract theorem on the existence of polynomial attractors and the concrete estimate of their attractive velocity for infinite-dimensional dynamical systems, then apply this theorem to a class of wave…

The Lorenz attractor is one of the best known examples of applied mathematics. However, much of what is known about it is a result of numerical calculations and not of mathematical analysis. As a step toward mathematical analysis, we allow…

We give the first examples of flows which exhibit robust singular attractors containing a wild hyperbolic set (in the sense of Newhouse). A hyperbolic set is said to be wild, if it has tangencies between its stable and unstable manifolds,…

We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to…

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which…

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. In [11] rate of convergence results in…

This paper is concerned with the homogenization of Dirichlet problem of elliptic systems in a bounded, smooth domain of finite type. Both the coefficients of the elliptic operator and the Dirichlet boundary data are assumed to be periodic…

We provide a systematic approach for deducing statistical limit laws via martingale-coboundary decomposition, for nonuniformly hyperbolic systems with slowly contracting and expanding directions. In particular, if the associated return time…

We consider a parabolic obstacle problem for Euler's elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably 'small'. For symmetric…

The density stratification in an incompressible fluid is responsible for the propagation of internal waves. In domains with topography, these waves exhibit interesting features. In particular, numerical and lab experiments show that, in two…

For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the…