The Dynamical Systems seminar is held on Monday afternoons at
4:00 PM in MCS 148. Tea
beforehand is at 3:45 PM in MCS 144.
January 23: Mike Todd (St Andrews)
Title: Stability of measures in interval dynamics
Abstract: Given a family of interval maps, each map possessing a `physical'
measure (an invariant measure absolutely continuous w.r.t. Lebesgue), we have
a weak form of stability if these measures change continuously through the
family. Even for uniformly hyperbolic dynamical systems this stability can
fail. I†¢ll give minimal conditions for a class of non-uniformly hyperbolic
interval maps to satisfy this stability property. This work forms part of a
paper with Neil Dobbs, where more general thermodynamic properties are proved
to be stable (entropy, pressure, equilibrium states), and I†¢ll give some
indication of the general approach there.
January 30: Eric Chang (BU)
Title: The Sierpinski Mandelbrot spiral for the rational map
Abstract: We investigate the parameter plane for the family of maps \(F(z) =
z^n + \lambda/z^d\) where \(n \geq 4\) is even, \(d \geq 3\) is odd, and
\(\lambda\) is a complex parameter. Concentrating on \(F(z) = z^4 + \lambda /
z^3\), we prove the existence of two structures in the parameter plane: a
Sierpinski Mandelbrot arc consisting of infinitely many alternating Sierpinski
holes and Mandelbrot sets, as well as a Sierpinski Mandelbrot spiral
consisting of infinitely many SM arcs. We also show that there are infinitely
many SM spirals in the parameter plane.
April 24: No seminar planned; please consider attending the workshop at
ICERM on Water Waves.
May 1: Maxim
Olshanii (UMass Boston)
Title: The Inverse Linearization Problem
Abstract: We investigate the relationship between the nonlinear
partial differential equations (PDEs) of mathematical physics and
the their linearizations around localized stationary solutions. It
turns out that for some classes of PDEs, it is possible to solve the
Inverse Linearization Problem, i.e. given the linearization, to restore
the original PDE. Of a particular interest are the instances of transparency
of the former that are shown to hint on the possible integrability of the