Project Title: Complex
Dynamics and Composition Operators
Adviser: Valentin Matache
Project Description: The field of complex dynamics designates the
behavior of complex selfmaps of a domain under
iteration, i.e. successive composition with themselves. The foundations of this
field were lain by P. Fatou
and R. Julia in a famous race for a Great Prize of the
The analytic selfmaps of the open unit disk make the object of a remarkable theorem proved by A. Denjoy and J. Wolff in 1926. The theorem asserts the existence of a remarkable point towards which the iterate sequence of such a map, (other than the identity or an elliptic disk automorphism) converge uniformly on compact subsets.
In the recent paper:
V. Matache, A fixed
Point Theorem for Analytic Functions, to appear in, Fixed Point Theory and
Applications,
the advisor proved a new interpolative fixed point theorem showing that analytic selfmaps of the unit disk simultaneously have fixed points if and only if they interpolate between select n-tuples. The main “power-tool” used in the proof of that result is a combination of results due to Denjoy, Julia, and Wolff.
The student interested in this project will be expected to follow research along this path as guided by the adviser. Among other things, he/she is expected to read the (very short!!!) paper above. Also, new results by the advisor and co-workers emphasize that the mode of convergence in the classical Denjoy-Wolff theorem can be significantly upgraded:
P. S. Bourdon, V.
Matache, and J.H. Shapiro, On Convergence of Sequences
of Iterates, submitted for publication.
The interested student will be helped by the advisor to read and understand this second paper and guided to do research about problems left open by the paper. In particular, (norm estimates for composition operators, i.e. transforms of analytic functions by composition with a fixed selfmap of the unit disk) will be of particular importance.
OTHER REQUIREMENTS: The students interested in the project above are
expected be enrolled
in MATH 4270(Complex Variables) during the Fall semester. They should be
familiar with computers and MAPLE, be willing to learn basic MATLAB (a CAS similar to MAPLE) and TeX, (a popular markup text-editor for mathematical papers)
by taking MATH 3500 (Intro to computer assisted mathematics research) in the
spring semester. The student is also expected to meet with the adviser a couple
of times a week, (or communicate actively by e-mail) for discussions, guidance,
and progress reports during the preparation period of the project.