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DCDS

This paper reviews some of the recent work of the author on stable
manifolds for unstable evolution equations. In particular, we
discuss such a result, jointly with Joachim Krieger, for the
critical focusing nonlinear wave equation in three dimensions.

JMD

We consider cocycles $\tilde A: \mathbb{T}\times K^d \ni (x,v)\mapsto (
x+\omega, A(x,E)v)$ with $\omega$ Diophantine, $K=\mathbb{R}$ or $K=\mathbb{C}$. We assume
that $A: \mathbb{T}\times \mathfrak{E} \to GL(d,K)$ is continuous, depends
analytically on $x\in\mathbb{T}$ and is Hölder in $E\in \mathfrak{E} $, where
$\mathfrak{E}$ is a compact metric space. It is shown that if all Lyapunov
exponents are distinct at one point $E_{0}\in\mathfrak{E}$, then they remain
distinct near $E$. Moreover, they depend in a Hölder fashion
on $E\in B$ for any ball $B\subset \mathfrak{E}$ where they are distinct.
Similar results, with a weaker modulus of continuity, hold for
higher-dimensional tori $\mathbb{T}^\nu$ with a Diophantine shift. We also
derive optimal statements about the rate of convergence of the
finite-scale Lyapunov exponents to their infinite-scale counterparts.
A key ingredient in our arguments is the Avalanche Principle, a
deterministic statement about long finite products of invertible
matrices, which goes back to work of Michael Goldstein and the
author. We also discuss applications of our techniques to products of
random matrices.

DCDS

In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions $3$ and $5$
assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made.
We find that there exist four sets in $\dot H^{1}\times L^{2}$ with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as $t → ±∞$.

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