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Spring Topology and Dynamical Systems Conference 2008
March 13-15, 2008
University of Wisconsin Milwaukee and Marquette University
Milwaukee, WI, USA

Organizers
Ric Ancel, Karen Brucks, Craig Guilbault, Chris Hruska, Suzanne Hruska, Boris Okun (UWM); Paul Bankston (Marquette); Lois Kailhofer (Alverno College).

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Monodromy of horseshoe Henon mappings and automorphisms of the two-shift
by
John Hamal Hubbard
Cornell University and Universite de Provence

Quadratic Henon mappings are written


Ha, c: (x, y) → (x2+c-ay, x).

For a ≠ 0 these are polynomial diffeomorphisms of C2.

Among these, in particular for |c| sufficiently large, one finds maps conjugate to a "Smale horseshoe" de Smale. Let X ⊂ C2 be the set of parameters for which the corresponding map is a horseshoe.

For x ∈ X, the set Kx ⊂ C2 of points (x, y) ∈ C2 with bounded orbit (both positive and negative) is conjugate to the full shift on two symbols. If x0 ∈ X is a base point, there is then a mondromy action


r: p1(X, x) → Aut(Kx).

The group of automorphisms of the full two shift is a pretty mysterious group. We will try to explain why various specific loops give rise to specific automorphisms.

Date received: February 29, 2008

Copyright © 2008 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 # cawy-36.