The school is designed for graduate students and
will use short lecture courses, tutorial and
discussion sessions, and student projects to
explore some of the fundamental concepts of
dynamical systems. It will be accessible
to students without a background in dynamics,
but is also intended for students who have begun
studying dynamics and wish to learn more about
this field.
The school is
funded by an NSF grant, which will cover
travel and local expenses for all accepted
participants. For
application instructions, see the bottom of
this page.
Short course topics
will include:
Decay of correlations in dynamical
systems (University of Houston
dynamics group: Vaughn Climenhaga, Matt
Nicol, Will Ott, Andrew Torok) These lectures will introduce the
notion of decay of correlations for a
dynamical system and will describe three
important methods for establishing a rate
of decay: spectral gap (Perron--Frobenius
theory); coupling techniques; and Birkhoff
cones.
Uniform hyperbolicity (Boris
Hasselblatt, Tufts University) Topological dynamics of hyperbolic
sets (shadowing, expansivity, closing,
spectral decomposition, specification,
topological stability, structural
stability, Markov approximation) from the
Anosov Shadowing Theorem. Ergodicity
and multiple mixing of product-like
measures (such as volume) from the Hopf
argument. Exotic contact Anosov flows by
Dehn--Foulon surgery. Optionally invariant
manifold theory or equilibrium states.
A diffeomorphism is partially hyperbolic
if there is an invariant splitting of the
tangent bundle into three subbundes, Es,
Ec, and Eu,
such that under iteration of the map
vectors in Es
shrink exponentially, vectors in Eu
expand exponentially and any
expansion/contraction of vectors in Ec
is weaker. This is a natural setting to
which many of the ideas from the study of
uniformly hyperbolic (Anosov) systems
extend. In particular Eberhard Hopf's
proof of ergodicity can be made to work
for a broad class of partially hyperbolic
systems. The lectures will give an
outline of this theory.
Rotations and interval exchange
transformations (Jon Chaika,
University of Utah)
Ergodicity and Weyl's criterion; Denjoy;
continued fraction algorithms to prove
Diophantine results via the Gauss map;
Sturmian sequences; weak mixing of 3-IETs;
some low complexity dynamics like bounds
on ergodic measures.
In addition to the mini-courses, there will
be a number of one-hour lectures introducing
various applications and extensions of the
theory presented in the short courses, such as
non-uniform hyperbolicity and Young towers;
applications to biology and neuroscience;
non-stationary systems; open systems;
martingales; statistical mechanics;
applications to models in finance.
Students participating
in the school should be familiar with the
following prerequisite material: measure theory,
basic functional analysis, basic theory of
smooth manifolds. There will be background
sessions during the school to give a brief
review of the most relevant parts of these
topics.
Funding for this event is provided by NSF
grant
DMS-1500151.
