Oudkerk, Richard (1999) The parabolic implosion for f0(z) = z + z v+1 + φ(2v=Z). PhD thesis, University of Warwick.

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Abstract

In this thesis
we examine the bifurcation in behaviour (for
the dynamics)
which occurs
when we perturb the holomorphic
germ
fo(z)
=z+ zv+1 +
O(zv+2) defined in
a neigh-
bourhood
of
0,
so that the
multiple
fixed
point at
0
splits
into
v+1
fixed
points
(counted
with multiplicity).
The
phenomenon observed is
called
the
parabolic
implosion,
since
the
perturbation will
typically
cause the filled Julia
set
(if it is defined) to "implode. "
The
main tool
used
for
studying
this bifurcation is the Fatou
coordinates and
the
associated
Ecalle
cylinders.
We
show the existence of
these for
a
family
of well
behaved
f's
close to fo,
and
that these depend continuously upon
f.
Each
well
behaved f
will
have
a gate structure which gives a qualitative
description
of the "egg-beater dynamic" for f. Each
gate
between the fixed
points of
f
will
have
an associated complex number called
the lifted
phase.
(Minus
the
real part of
the lifted
phase is
a rough measure of
how
many
iterations it takes
for
an orbit to
pass through the
gate. ) The
existence of maps with any
desired
gate structure and any
(sensible) lifted
phases is
shown.
This leads to
a simple parameterisation of
the
well
behaved maps.
We
are particularly
interested in
sequences fk
→
fo
where all
the lifted
phases of the
fk
converge to
some
limits,
modulo
Z. We
show that there is
a non-trivial
Lavaurs
reap
g associated with these limits,
which commutes with
fo. The dynamics
of
fk
are shown
to (in
some sense) converge to the dynamics
of
the
system
(fo,
g).
We
also prove
that for
any
Lavaurs map g there is
a sequence fk
→
fo
so that
fk
k
→ g
as
k
→ +oo, uniformly on compact sets.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Dynamics, Bifurcation theory
Official Date:	June 1999
Dates:	Date Event June 1999 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Lei, Tan
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC)
Extent:	vii, 94 leaves : illustrations
Language:	eng

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