First workshop of the Warwick Dynamics Symposium 2002-2003 (December 9th-13rd)
The main theme of the December
workshop is the interface between real and complex dynamics.
Programme with Abstracts (Up to date as of 6th of December 4.30pm.)
Registered Particpants
(Up to date as of 5th of December.)
One dimensional dynamics is a
very active research area worldwide.
The recent developments in these fields show that a combination of real and
complex tools and methods might be
very successful. The original proofs of exponential decay of
geometry is mostly complex, but these results can be extended
to the unimodal case. On the other hand, many notions
and ideas of complex dynamics (like
renormalization, Collet-Eckmann condition, summability condition, etc) are
coming from real dynamics. There are
many other examples of this kind when achievement in one branch of 1D
dynamics immediately lead to success in the
other one. We intend to bring together people working in both real and
complex theories and organize an intensive
interchange of new results and techniques between them.
A few of the topics which will come up during the meeting are:
Density of Axiom A maps.
The structural stability problem was and remains
one of the central problems in the
theory of Dynamical systems. The case of real quadratic (and more generally
unimodal) maps is done (by Graczyk,
Kozlovski, Lyubich, Sullivan, Swiatek, Thurston, Yoccoz. However only
partial results are known for multimodal maps.
For complex polynomials the same question is wide open and the main result
here remains the Yoccoz theorem for finitely renormalizable quadratic maps.
Absence of invariant line field and measure of the Julia set.
This topic is
connected to the previous one. Indeed, Mañè,
Sad and Sullivan in their famous paper have shown that if a complex
polynomial is not Axiom A and cannot be
perturbed by a small perturbation to an Axiom A map, then such a polynomial
admits a measurable invariant line
field of its Julia set (in particular, the Julia set of such a polynomial
has measure zero). So, to prove the density of
Axiom A for complex polynomials (or rational maps) one has at least to
prove the absence of invariant line fields.
For real polynomials this was done by Levin and van Strien for real
unicritical polynomials and recently by Shen (who
is presently at Warwick) for arbitrary real polynomials. Unfortunately
these results do not yet give much information
for perturbation of real non-unimodal polynomials.
SRB measures.
The recent work of Lyubich, Avila and de Melo, for families
of unimodal maps with a quadratic critical
point. These results heavily use specific properties of quadratic maps, so
generalizations for multimodal families and
unimodal families of maps having a degenerate critical point are certainly
not straightforward. Some other results in
this direction are obtained by Smirnov, Graczyk and Swiatek for complex
quadratic polynomials.
Henon maps.
During this session a few experts will discuss aspects of Benedicks and
Carleson's paper about the strange attractors in Henon maps.
This meeting will be organized by Oleg Kozlovski, Weixiao Shen and Sebastian
van Strien.