Abstracts of the talks at the
Ph.d. course and workshop
Holomorphic Dynamics
- Iterated Monodromy groups and Henon maps with a semi-neutral fixed point -
Søminestationen Holbæk,
November 30. – December 3. 2017
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The backbone of the combined workshop and ph.d course wil be two mini courses one on Iterated Monodromy groups held by Laurent Bartholdy and Russel Lodge and one on Henon maps with a semi-neutral fixed point by Remus Radu and Raluca Tanase. These mini-courses are supplemented with contributed talks on holomorphic dynamics in general. |
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Mini-course Title : Iterated Monodromi Groups. Bartholdi
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-------------------------------------------------------------- I will talk about
the general idea of using an algebraic invariant, the biset, to study maps up to homotopy.
Talk 2: Contracting IMGs,
hyperbolic bisets, and expanding maps. --------------------------------------------------------------------------------------- The connection between groups
and dynamics is sharper in case the IMG has the "contraction"
property. I will talk about the connection between expansion of a map
and contraction of the biset. Nekrashevych's construction of the limit space of a biset.
Abstract G-spaces. Dudko's theorem: a sphere map is
isotopic to an expanding map if and only if its biset
is contracting, if and only if it has no Levy cycle. Encoding by automata. Talk 3: Decompositions of
maps and bisets. ---------------------------------------------------------- Bisets can be decomposed as "amalgamated free
products", just as spheres can be cut along annuli and Thurston maps can
be decomposed along obstructions. I will show how the algebraic and dynamic
decompositions correspond to each other. Canonical decompositions of bisets. Graphs of spheres = graphs of groups with
infinite cyclic groups on the edges. Graphs of maps, given by a
correspondence and a multicurve = graphs of bisets with cyclic bisets on
the edges. Canonical obstructions to expansion and to algebraicity. Talk 4: The conjugacy
problem in bisets. ----------------------------------------------------- The "Thurston
equivalence" problem can be expressed as conjugacy in a mapping class biset. I will explain how mapping class bisets generalize mapping class groups, and how bisets can be used to solve algorithmically the question
of whether two maps are Thurston equivalent. The mapping class biset of a map = its orbit under composition, left and
right, by mapping classes. Thurston equivalence of Thurston maps = conjugacy
class in the mapping class biset. Decidability, by
decomposing, computing canonical obstructions (first to expansion, then to algebraicity). Examples of calculation of mapping class bisets. |
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Lodge: Talk 1. Thurston's theorem: applications and proof. -------------------------------------------------------------- W. Thurston's characterization and rigidity theorem for postcritically finite rational maps has fundamental importance in the combinatorial study of holomorphic dynamical systems. For years, applications of Thurston's theorem were limited by the lack of efficient computational tools, a problem that has recently been addressed by the theory of self-similar groups. I will present some applications of Thurston's theorem, and discuss the proof using iteration of the pullback map on Teichmuller space. The pullback map has further implications for the group theory, and can be used to check the hypothesis of Thurston's theorem in some cases that I will describe. Talk 2. Polynomial bisets and
the combinatorial spider algorithm. -------------------------------------------------------------- A rich combinatorial theory for postcritically finite polynomials has existed for years, relying on the fact that the basin of infinity is totally invariant. Using certain invariant structures in this basin, I will give the classification of bisets associated to polynomials and show how they can be used in conjunction with the "combinatorial spider algorithm" to determine Thurston class in the presence of certain algebraic properties. Mini-course |
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Title : Henon maps with a
semi-neutral fixed point. Abstract: Complex Henon maps are a special case
of polynomial automorphisms of C^2 and are central
objects in the study of holomorphic dynamics in 2D. In this mini-course, we
give a unified treatment on Henon maps with a
semi-neutral fixed point (i.e. which has one eigenvalue of absolute value one
and one eigenvalue of absolute value less than one). We describe the local
and in some cases even global dynamics of these non-hyperbolic maps. As in
dimension one, we can distinguish three cases: semi-parabolic, semi-Siegel
and semi-Cremer. We outline the different behavior in
each case, discuss recent progress, and make analogies to one-dimensional
dynamics. Lecture 1: Local and global dynamics of semi-parabolic Henon maps Lecture 2: Stability and continuity of Julia sets in C2 Lecture 3: Siegel disks for complex Henon maps Lecture 4: Hedgehogs in higher dimensions and their applications to Henon maps. |
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Title : Modular correspondences. |
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Title : Julia sets with wandering branching points. |
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Title : Iterated Monodromy Groups of Entire
Functions. |
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