Generic dynamics of 4-dimensional C2 Hamiltonian systems

We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C2-residual set of Hamiltonians for which every regular energy surface is either Anosov or it is in the closure of energy surfaces with zero Lyapunov exponents a.e. This is in the spirit of the Bochi-Mane dichotomy for area-preserving diffeomorphisms on compact surfaces and its continuous-time version for 3-dimensional volume-preserving flows

LpL^p-generic cocycles have one-point Lyapunov spectrum

We show the sum of the first $k$ Lyapunov exponents of linear cocycles is an upper semicontinuous function in the $L^p$ topologies, for any $1 \le p \le \infty$ and $k$. This fact, together with a result from Arnold and Cong, implies that the Lyapunov exponents of the $L^p$-generic cocycle, $p<\infty$, are all equal.Comment: 8 pages. A gap in the previous version was correcte

Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schr\"odinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schr\"odinger operator is a Cantor set.Comment: Final version. To appear in Duke Mathematical Journa

Hyperbolicity and the effective dimension of spatially-extended dissipative systems

We show, using covariant Lyapunov vectors, that the chaotic solutions of spatially extended dissipative systems evolve within a manifold spanned by a finite number of physical modes hyperbolically isolated from a set of residual degrees of freedom, themselves individually isolated from each other. In the context of dissipative partial differential equations, our results imply that a faithful numerical integration needs to incorporate at least all physical modes and that increasing the resolution merely increases the number of isolated modes.Comment: 4 pages, 4 figure

Dominated Splitting and Pesin's Entropy Formula

Let $M$ be a compact manifold and $f:\,M\to M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting $T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$, then we give an estimation through Lyapunov characteristic exponents from below in Pesin's entropy formula, i.e., the metric entropy $h_\mu(f)$ satisfies $$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ where $\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and $\lambda_1(x)\geq\lambda_2(x)\geq...\geq\lambda_{dim\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $\mu.$ Consequently, by using a dichotomy for generic volume-preserving diffeomorphism we show that Pesin's entropy formula holds for generic volume-preserving diffeomorphisms, which generalizes a result of Tahzibi in dimension 2

Some open questions in "wave chaos"

The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more recent

Magnetoelastic mechanism of spin-reorientation transitions at step-edges

The symmetry-induced magnetic anisotropy due to monoatomic steps at strained Ni films is determined using results of first - principles relativistic full-potential linearized augmented plane wave (FLAPW) calculations and an analogy with the N\'eel model. We show that there is a magnetoelastic anisotropy contribution to the uniaxial magnetic anisotropy energy in the vicinal plane of a stepped surface. In addition to the known spin-direction reorientation transition at a flat Ni/Cu(001) surface, we propose a spin-direction reorientation transition in the vicinal plane for a stepped Ni/Cu surface due to the magnetoelastic anisotropy. We show that with an increase of Ni film thickness, the magnetization in the vicinal plane turns perpendicular to the step edge at a critical thickness calculated to be in the range of 16-24 Ni layers for the Ni/Cu(1,1,13) stepped surface.Comment: Accepted for publication in Phys. Rev.

Absence of stable collinear configurations in Ni(001)ultrathin films: canted domain structure as ground state

Brillouin light scattering (BLS) measurements were performed for (17-120) Angstrom thick Cu/Ni/Cu/Si(001) films. A monotonic dependence of the frequency of the uniform mode on an in-plane magnetic field H was observed both on increasing and on decreasing H in the range (2-14) kOe, suggesting the absence of a metastable collinear perpendicular ground state. Further investigation by magneto-optical vector magnetometry (MOKE-VM) in an unconventional canted-field geometry provided evidence for a domain structure where the magnetization is canted with respect to the perpendicular to the film. Spin wave calculations confirm the absence of stable collinear configurations.Comment: 6 pages, 3 figures (text, appendix and 1 figure added

Dynamics of Excited Electrons in Copper and Ferromagnetic Transition Metals: Theory and Experiment

Both theoretical and experimental results for the dynamics of photoexcited electrons at surfaces of Cu and the ferromagnetic transition metals Fe, Co, and Ni are presented. A model for the dynamics of excited electrons is developed, which is based on the Boltzmann equation and includes effects of photoexcitation, electron-electron scattering, secondary electrons (cascade and Auger electrons), and transport of excited carriers out of the detection region. From this we determine the time-resolved two-photon photoemission (TR-2PPE). Thus a direct comparison of calculated relaxation times with experimental results by means of TR-2PPE becomes possible. The comparison indicates that the magnitudes of the spin-averaged relaxation time \tau and of the ratio \tau_\uparrow/\tau_\downarrow of majority and minority relaxation times for the different ferromagnetic transition metals result not only from density-of-states effects, but also from different Coulomb matrix elements M. Taking M_Fe > M_Cu > M_Ni = M_Co we get reasonable agreement with experiments.Comment: 23 pages, 11 figures, added a figure and an appendix, updated reference