Student/Faculty Colloquium-September 9, 1998
Department of Mathematics
Texas A & M - Commerce
Student/Faculty Colloquium
Wednesday, September 9, 1998
3:00-4:00pm, Bin-329
David Delatte, of the
University of North Texas,
will speak on:
Small divisors and overdetermined systems
Abstract:
A technique that overcomes small divisor problems in
overdetermed systems of functional equations will be discussed. Our
focus will be on a linearization theorem for commuting holomorphic
functions. The proof involves KAM theory for overdetermined systems
subject to a very general number theoretic (diophantine) condition on
the small divisors.
More precisely:
Let f,g: C-->C with a common fixed point at the origin
and suppose that f(z) = lambda*z + O(z2), g(z) = gamma*z +
O(z2), and neither lambda nor gamma are zero.
The map, f is called
linearizable if there is an analytic diffeomorphism, h, which
conjugates f with its linear part in a neighborhood of the origin,
i.e., h-1 o f o h (z) = lambda*z where lambda =
f'(0).
Two such diffeomorphisms are simultaneously linearizable
if they are linearized by the same map, h. Our main theorem uses a
diophantine condition on the pair lambda, gamma that ensures the
simultaneous linearizability of the commuting holomorphic functions
f and g.
OUTLINE:
-
Just how irrational is that number?
-
Rotating tires, circles, complex numbers.. ahh it's all the same.
- When do two dynamical systems look the same (in a neighborhood)?
- Commuting systems and symmetry
- Small divisors: what they are, how to avoid them and what to do
when you can't.
- Trading real estate for estimates: Application of KAM theory to an
overdetermined sytem.