39,748 research outputs found
Dessins d'enfants and Hubbard Trees
We show that the absolute Galois group acts faithfully on the set of Hubbard
trees. Hubbard trees are finite planar trees, equipped with self-maps, which
classify postcritically finite polynomials as holomorphic dynamical systems on
the complex plane. We establish an explicit relationship between certain
Hubbard trees and the trees known as ``dessins d'enfant'' introduced by
Grothendieck.Comment: 27 pages, 8 PostScript figure
The realization of input-output maps using bialgebras
The theory of bialgebras is used to prove a state space realization theorem for input/output maps of dynamical systems. This approach allows for the consideration of the classical results of Fliess and more recent results on realizations involving families of trees. Two examples of applications of the theorum are given
Approximability of dynamical systems between trees of spheres
We study sequences of analytic conjugacy classes of rational maps which diverge in moduli space. In particular, we are interested in the notion of rescaling limits introduced by Jan Kiwi. In the continuity of [A1] we recall the notion of dynamical covers between trees of spheres for which a periodic sphere corresponds to a rescaling limit. We study necessary conditions for such a dynamical cover to be the limit of dynamically marked rational maps. With these conditions we classify them for the case of bicritical maps and we recover the second main result of Jan Kiwi regarding rescaling limits
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