Job Description |
Newton's method for entire functions (task D2)
Newton's method is a well-known algorithm of finding the zeros of a
differentiable function, and its study has played a key role in the
development of complex dynamics. For polynomials important results
about the size and geometry of the basins of attractions have been
obtained. E.g., Hubbard, Schleicher and Sutherland have shown that for
a polynomial of degree d one needs at most 1.11 d (log d)^2 starting
points to find all zeros. Also, the (spherical) area of the basins of
attractions is bounded by c/d for some absolute constand c, provided
the polynomial is suitably normalized. There are also some results
about
Newton's method for entire functions. E.g., Bergweiler and Terglane
have shown that Sullivan's no wandering domains theorem extends to
Newton's method for certain classes of entire functions, and Jankowski
has considered the size and geometry of the attracting basins in some
special cases. But in general much less is known about Newton's method
for entire functions. The purpose of this project is to study Newton's
method for much more general classes of entire functions, and to study
it in greater detail for some of the classes already considered. In
particular, it would be of interest to find "small" sets of starting
values that find all roots. Also, it seems reasonable that the area
(in the plane) of the attracting basins is infinite for functions of
small order of growth. Difficulties arise in particular if the
restriction of the Newton map to the immediate attracting basin is
not a proper map.
The task will be performed in Kiel (Germany) by a PhD student
appointed
as ESR. There will be regular contact in particular with
Bremen and with the Spanish team.
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