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Series: CDSNS Colloquium

Consider a generic perturbation of a nearly integrable system
of {\it arbitrary degrees of freedom $n\ge 2$ system}\[H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T,\]with strictly convex $H_0$. Jointly with P.Bernard and K.Zhang we prove existence of orbits $(\th,p)(t)$ exhibiting Arnold diffusion
\[\|p(t)-p(0) \| >l(H_1)>0 \quad \textup{independently of }\eps.\]Action increment is independent of size of perturbation$\eps$, but does depend on a perturbation $\eps H_1$.This establishes a weak form of Arnold diffusion.
The main difficulty in getting rid of $l(H_1)$ is presence of strong double resonances. In this case for $n=2$we prove existence of normally hyperbolic invariant manifolds passing through these double resonances. (joint with P. Bernard and K. Zhang)

Series: CDSNS Colloquium

I will discuss recent work on the stability of linear
equations under
parametric forcing by colored noise. The noises considered are built from
Ornstein-Uhlenbeck vector processes. Stability of the solutions is
determined by the boundedness of their second moments. Our
approach uses the Fokker-Planck equation and the associated PDE
for the marginal moments to determine the growth rate of the
moments. This leads to an eigenvalue problem, which is solved
using a decomposition of the Fokker-Planck operator for Ornstein-Uhlenbeck processes
into "ladder operators." The results are given in terms of a perturbation
expansion in the size of the noise. We have found very good
agreement between our results and numerical simulations. This is
joint work with L.A. Romero.

Series: CDSNS Colloquium

An important question in circle dynamics is regarding the absolute continuity of aninvariant measure. We will consider orientation preserving circle homeomorphisms withbreak points, that is, maps that are smooth everywhere except for several singular pointsat which the rst derivative has a jump. It is well known that the invariant measuresof sufficiently smooth circle dieomorphisms are absolutely continuous w.r.t. Lebesguemeasure. But in the case of homeomorphisms with break points the results are quitedierent. We will discuss conjugacies between two circle homeomorphisms with breakpoints.Consider the class of circle homeomorphisms with one break point b and satisfying theKatznelson-Ornsteins smoothness condition i.e. Df is absolutely continuous on [b; b + 1]and D2f 2 Lp(S1; dl); p > 1: We will formulate some results concerning the renormaliza-tion behavior of such circle maps.

Series: CDSNS Colloquium

The discrete Schrodinger operator with Fibonacci potential is a central model in the study of electronic properties of one-dimensional quasicrystals. Certain renormalization procedure allows to reduce many questions on specral properties of this operator to the questions on dynamical properties of a so called trace map. It turnes out that the trace map is hyperbolic, and its measure of maximal entropy is directly related to the integrated density of states of the Fibonacci Hamiltonian. In particular, this provides the first example of an ergodic family of Schrodinger operators with singular density of states measure for which exact dimensionality can be shown. This is a joint work with D. Damanik.

Series: CDSNS Colloquium

With recent advances in experimental imaging, computational methods,
and dynamics insights it is now possible to start charting out the
terra incognita explored by turbulence in strongly nonlinear classical
field theories, such as fluid flows. In presence of continuous
symmetries these solutions sweep out 2- and higher-dimensional
manifolds (group orbits) of physically equivalent states,
interconnected by a web of still higher-dimensional stable/unstable
manifolds, all embedded in the PDE infinite-dimensional state spaces.
In order to chart such invariant manifolds, one must first quotient the
symmetries, i.e. replace the dynamics on M by an equivalent, symmetry
reduced flow on M/G, in which each group orbit of symmetry-related
states is replaced by a single representative.Happy news: The
problem has been solved often, first by Jacobi (1846), then by Hilbert
and Weyl (1921), then by Cartan (1924), then by [...], then in this
week's arXiv [...]. Turns out, it's not as easy as it looks.Still,
every unhappy family is unhappy in its own way: The Hilbert's solution
(invariant polynomial bases) is useless. The way we do this in quantum
field theory (gauge fixing) is not right either. Currently only the
"method of slices" does the job: it slices the state space by a set of
hyperplanes in such a way that each group orbit manifold of
symmetry-equivalent points is represented by a single point, but as
slices are only local, tangent charts, an atlas comprised from a set of
charts is needed to capture the flow globally. Lots of work and not a
pretty sight (Nature does not like symmetries), but one is rewarded by
much deeper insights into turbulent dynamics; without this atlas you
will not get anywhere.This is not a fluid dynamics talk. If you
care about atomic, nuclear or celestial physics, general relativity or
quantum field theory you might be interested and perhaps help us do
this better.You can take part in this seminar from wherever you are by clicking onevo.caltech.edu/evoNext/koala.jnlp?meeting=M2MvMB2M2IDsDs9I9lDM92

Series: CDSNS Colloquium

Hyperbolic actions of Z^k and R^k arise naturally in algebraic
and geometric context. Algebraic examples include actions by
commuting automorphisms of tori or nilmanifolds and, more generally,
affine and homogeneous actions on cosets of Lie groups. In contrast
to hyperbolic actions of Z and R, i.e. Anosov diffeomorphisms and
flows, higher rank actions exhibit remarkable rigidity properties,
such as scarcity of invariant measures and smooth conjugacy to
a small perturbation. I will give an overview of results in this area
and discuss recent progress.

Series: CDSNS Colloquium

A linear cocycle over a diffeomorphism f of a manifold M is
an automorphism of a vector bundle over M that projects to f.
An important example is given by the differential Df or its
restriction to an invariant sub-bundle. We consider a Holder
continuous linear cocycle over a hyperbolic system and explore
what conclusions can be made based on its properties at the
periodic points of f. In particular, we obtain criteria for
a cocycle to be isometric or conformal and discuss applications
and further developments.

Series: CDSNS Colloquium

Consider a hyperbolic basic set of a smooth diffeomorphism. We are
interested in the transitivity of Holder skew-extensions with fiber a
non-compact connected Lie group.
In the case of compact fibers, the transitive extensions contain an open
and dense set. For the non-compact case, we conjectured that this is still
true within the set of extensions that avoid the obvious obstructions to
transitivity. Within this class of cocycles, we proved generic transitivity
for extensions with fiber the special Euclidean group SE(2n+1) (the case
SE(2n) was known earlier), general Euclidean-type groups, and some
nilpotent groups.
We will discuss the "correct" result for extensions by the Heisenberg
group: if the induced extension into its abelinization is transitive, then
so is the original extension. Based on earlier results, this implies the
conjecture for Heisenberg groups. The results for nilpotent groups involve
questions about Diophantine approximations.
This is joint work with Ian Melbourne and Viorel Nitica.

Series: CDSNS Colloquium

The study of transport is an active area of applied mathematics of interest to fluid mechanics, plasma physics, geophysics, engineering, and biology among other areas. A considerable amount of work has been done in the context of diffusion models in which, according to the Fourier-‐Fick’s prescription, the flux is assumed to depend on the instantaneous, local spatial gradient of the transported field. However, despiteits relative success, experimental, numerical, and theoretical results indicate that the diffusion paradigm fails to apply in the case of anomalous transport. Following an overview of anomalous transport we present an alternative(non-‐diffusive) class of models in which the flux and the gradient are related non-‐locally through integro-differential operators, of which fractional Laplacians are a particularly important special case. We discuss the statistical foundations of these models in the context of generalized random walks with memory (modeling non-‐locality in time) and jump statistics corresponding to general Levy processes (modeling non-‐locality in space). We discuss several applications including: (i) Turbulent transport in the presence of coherent structures; (ii) chaotic transport in rapidly rotating fluids; (iii) non-‐local fast heat transport in high temperature plasmas; (iv) front acceleration in the non-‐local Fisher-‐Kolmogorov equation, and (v) non-‐Gaussian fluctuation-‐driven transport in the non-‐local Fokker-‐Planck equation.

Series: CDSNS Colloquium

We generalize some notions that have played an important
role in dynamics, namely invariant manifolds, to the
more general context of difference equations. In particular,
we study Lagrangian systems in discrete time. We define
invariant manifolds, even if the corresponding difference
equations can not be transformed in a dynamical system.
The results apply to several examples in the Physics literature:
the Frenkel-Kontorova model with long-range interactions
and the Heisenberg model of spin chains with a
perturbation. We use a modification of the parametrization
method to show the existence of Lagrangian stable
manifolds. This method also leads to efficient algorithms
that we present with their implementations.
(Joint work with Rafael de la Llave.)