Related papers: Proof of Wojtkowski's Falling Particle Conjecture
We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…
This is an introduction of a book called "strong regularity", to appear at Ast\'erisque, containing: 1) Yoccoz' proof of Jakobson theorem www.college-de-france.fr/media/jean-christophe-yoccoz/UPL7416254474776698194_Jakobson_jcy.pdf 2)…
We consider the motion of several rigid bodies immersed in a two-dimensional incompress-ible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler…
We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.
An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified…
The theoretical cornerstone of statistical mechanics is the ergodic assumption that all accessible configurations of a physical system are equally likely. Here we show how such property arises when an open quantum system is continuously…
In this paper, we study diagonalizable hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and nondecreasing initial data. Moreover, we show…
In 1998 Kleinbock conjectured that any set of weighted badly approximable $d\times n$ real matrices is a winning subset in the sense of Schmidt's game. In this paper we prove this conjecture in full for vectors in $\mathbf{R}^d$ in…
Recently, Kiriki, Nakano and Soma introduced a concept called pointwise emergence as a new quantitative perspective into the study of non-existence of averages for dynamical systems. In the present paper, we consider the set of points with…
We revisit a method introduced by Tartar for proving global well-posedness of a semilinear hyperbolic system with null quadratic source in one space dimension. A remarkable point is that, since no dispersion effect is available for 1D…
We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.
Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemanczyk, De La Rue and by the observation that the Mobius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem in…
We obtain a system for the spatial metric and extrinsic curvature of a spacelike slice that is hyperbolic non-strict in the sense of Leray and Ohya and is equivalent to the Einstein equations. Its characteristics are the light cone and the…
It is known that a gambler repeating a game with positive expected value has a positive probability to never go broke. We use the mass transport method to prove the generalization of this fact where the gains from the bets form a…
This paper investigates the ergodicity of Markov--Feller semigroups on Polish spaces, focusing on very weak regularity conditions, particularly the Ces\`aro eventual continuity. First, it is showed that the Ces\`aro average of such…
This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two…
This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincar\'e…
The theory presented in this monograph establishes the first mathematically rigorous result on the global nonlinear stability of self-gravitating matter under small perturbations of an asymptotically flat, spacelike hypersurface of…
In this paper, we present and analyze a weak Galerkin finite element (WG) method for solving the symmetric hyperbolic systems. This method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary…
It is well known that ergodic theory can be used to formally prove a weak form of relaxation to equilibrium for finite, mixing, Hamiltonian systems. In this Letter we extend this proof to any dynamics that preserves a mixing equilibrium…