Advanced
Writing Projects.
For
Students of the Advanced Dynamics
Course,
possibly for Master's Theses' Projects.
Students
should write 10 to 20 pages (in Latex)
of
expository mathematics on one of the
following
topics.
(For
references on these problems,
please,
consult me.)
Thurston's
Algorithm and monotonicity of the kneading sequence.
Continued
fractions and quadratic forms: the Markoff spectrum through scalings.
Continued
fractions and hyperbolic geometry.
Continued
fractions and number theory: (Liouville -, Diophantine -, and Roth numbers;
the
growth rate of the continued fraction coefficients).
The
Hartmann-Grobman Theorem (and extensions).
The
smoothness estimate of Herman's Theorem.
Ergodic
Theory.
Smooth
Expanding Maps on compact manifolds.
Sarkovski's
Theorem.
Define
and discuss: Invariant sets (backward and forward),
invariant
measures (forward and backward), alpha and omega limit sets, attractors,
non-wandering
sets, basin of an attractor, stable and unstable manifolds.
Iterated
functions systems, their actions on sets and on probability measures.
Minimally
separating sets in dynamical systems (Lakes of Wada, Birkhoff attractors,
etc).
Non-linearity
estimates in 1 dimension and in dimension greater than 1.
Laplacians
on lattices and graphs: Stokes's Theorem, expected return time, expected
covering time,
Markov
chains, random walks, etc.
Existence,
uniqueness, and smoothness of solutions of
smooth,
ordinary differentialequations.