Residual Julia Sets of Meromorphic
Functions
Dr. Tuen Wai Ng
Abstract
Given a meromorphic function, the complex plane can then be
partitioned into two parts, its Fatou set and Julia
set. The residual Julia set of a meromorphic function
is defined to be the subset of the Julia set that contains points which are not
the boundary points of any Fatou component. In 1988,
C. McMullen proved the surprising result that there exist some rational
functions whose residual Julia sets are non-empty. In 1990, Makienko
conjectured that for any rational function of degree greater than one, its residual
Julia set is empty if and only if the Fatou set has a
completely invariant component or consists of only two components. Since then the
conjecture has been verified for several classes of rational functions.
However, the conjecture remains open.
In this
talk, we shall show that the conjecture is true for any meromorphic
function with only finitely many critical values and asymptotic values and its
Julia set is locally connected. This result covers all previous known results
on Makienko's conjecture. We shall also prove certain
properties of residual Julia sets which generalizes a
result of Baker and Domínguez. This is a joint work with J.H. Zheng
and Y.Y. Choi.
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Date: |
November 25, 2004 (Thursday) |
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Time: |
4:00 – 5:00pm |
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Place: |
Room 517, Meng Wah Complex |
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All are welcome |