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Conformal measures and limit sets
by
Lex G. Oversteegen
UAB
Coauthors: A. Blokh, J. Mayer
Lex G. Oversteegen
We will study rational maps of the complex sphere. We are interested
in studying the limit behavior of iterations of almost''
every point
of the sphere under iteration of an arbitrary rational map.
For points in the Fatou set this follows from Sullivan's
Classification
Theorem of Fatou components. In this talk we will focus on
the behavior
of points in the Julia set J. In case the Lebesgue measure m
of J is positive,
we will relate the omega limit set of m-almost every point in J
to the omega
limit set of the recurrent critical points. Otherwise it is
known that
there exists a conformal measure \mu with support on J.
We will
generalize the classical Lebesgue Density Theorem to conformal
measures and
conformal balls of limited distortion and use this to relate
the omega limit set of
\mu-almost every point in J to the omega limit set
of the critical points.
These results are joint work with A. Blokh and J. Mayer.
Date received: February 16, 1998
Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caas-88.