On Yoccoz Return Functions

Details On Yoccoz Return Functions On Yoccoz Return Functions

2009-08-23

arxiv.org/abs/0908.3326v1

Mathematics

Dynamical Systems

33 pages, 3 figures

Scientific paper

We study the dynamics of complex polynomials. We obtain results on Poincare return maps defined on certain neighborhoods of a point with bounded orbit under a polynomial. We introduce a generalization of the Yoccoz tau-function, the Yoccoz return function, which codes the returns of a critical point with bounded orbit of any complex polynomial with a disconnect Julia set. We give necessary conditions on Yoccoz return functions, which allow for the recursive definition of an abstract tau-function. These conditions are also sufficient for polynomials that have a disconnected Julia set and exactly one critical point with bounded orbit.

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