In this paper, we consider a time independent $C^2$ Hamiltonian, sa\-tisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.
In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.
For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$, we construct a unique forward weak KAM solution of \[ H(x, d_x u)=c(H) \] by a vanishing discount approach, where $c(H)$ is the Mañé critical value. We also discuss the dynamical significance of such a special solution.
We revisit non-autonomous systems depending quasi-periodically in time within the reversible context 2 of KAM theory and obtain Whitney smooth families of invariant tori in such systems via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question.
In this paper, we study a class of time-periodic stochastic Tonelli Lagrangians on compact manifolds. Precisely, we discuss the stochastic Mane Critical Value, prove the existence of stochastic Weak KAM solutions of the related Hamilton-Jacobi equation. Furthermore, we survey the global minimizer.
We prove a general theorem on the persistence of Whitney infinitely smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim Fix G < (codim T)/2 where Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H.W.Broer, M.-C.Ciocci, H.Hanssmann, and A.Vanderbauwhede of 2009 concerning quasi-periodic stability of invariant tori with singular "normal" matrices in reversible systems.
Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.
We consider the Hamilton-Jacobi equation \[{H}(x,u,Du)=0,\quad x\in M, \] where $M$ is a connected, closed and smooth Riemannian manifold, ${H}(x,u,p)$ satisfies Tonelli conditions with respect to $p$ and certain decreasing condition with respect to $u$. Based on a dynamical approach developed in \cite{WWY,WWY1,WWY2}, we obtain a series of properties for weak KAM solutions (equivalently, viscosity solutions) of the stationary equation and the long time behavior of viscosity solutions of the evolutionary equation on the Cauchy problem \begin{equation*} \begin{cases} w_t+{H}(x,w,w_x)=0,\quad (x,…
Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case
Universitat de Barcelona（巴塞罗那大学）；Centre de Mathématiques Laurent Schwartz (CMLS), CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院中央数学实验室）
We consider the equation $Δ_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We prove that the equation possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in $x'=(x_1,\dots,x_{N-1})$ and decaying as $|x'|\to\infty$, periodic in $x_N$, and quasiperiodic in $y$. Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type…
This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $0<\frac{\partial H}{\partial u}\leqslant λ$ for some $λ>0$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry s…
The Lax-Oleinik semi-group: a Hamiltonian point of view
弱KAM理论是由Fathi为了研究凸哈密顿系统的动力学而发展起来的。它在某种程度上使哈密顿-雅可比方程的黏性解和哈密顿系统的马瑟不变集之间架起了一座桥梁，尽管这只是一个后验的理解。这些理论在凸性假设下趋于一致，应用的丰富性主要来自于这种显著的一致性。在本课程中，我们提供了弱KAM理论的一些基本概念的初步阐述。在一个同伴讲座，阿尔伯特法特希暴露方面，他的理论是更直接相关的黏性解。相反，这里我们关注的是动力学应用，即使我们也讨论了一些粘性方面，以强调与Fathi讲座的联系。关于弱KAM理论的基本参考是Albert Fathi \textit{Weak KAM theorem in Lagrangian dynamics}尚未出版的书。虽然我们没有提供新的结果，但我们的论述在几个方面是原创的。我们只使用哈密顿量，不依赖拉格朗日量，即使有些证明直接来自经典拉格朗日证明。通过选择某种特定的设置，这种方法变得更容易。我们研究了$\Rm^d$，并对哈密顿量作了统一假设。这使得我们可以用明确的估计来代替一些紧性参数。然而，对于最有趣的动力学应用程序，紧凑的配置空间仍然是一个有用的假设，我们检索它考虑周期（在空间中）哈密顿。本文主要讨论Hamilton-Jacobi方程的Cauchy问题和与之相关的Lax-Oleinik发展算子，并通过考虑这些发展算子的不动点，即弱KAM解，得到动力学应用.演化算子…
The paper contains an application of van Kampen theorem for groupoids for computation of homotopy types of certain class of non-compact foliated surfaces obtained by gluing at most countably many strips $\mathbb{R}\times(0,1)$ with boundary intervals in $\mathbb{R}\times\{\pm1\}$ along some of those intervals.
We study smooth foliations of arbitrary codimension on homogeneous compact Kähler manifolds. We prove that smooth foliations on rational compact homogeneous manifolds are locally trivial fibrations and classify the smooth foliations with all leaves analytically dense on compact homogeneous Kähler manifolds. Both results are builded upon a (rough) structure Theorem for smooth foliations on compact homogeneous Kähler manifolds obtained by comparison of the foliation and the Borel-Remmert decomposition of the ambient.
In this note we formulate a condition for complete, connected and non-compact Riemannian manifolds which implies no conjugate points in case that the geodesic flow is Anosov with respect to the Sasaki metric.
