| Résume | Skew-products are maps of the form f(x,y)=(P(x),Q(x,y)). We are intereste on the situation where P=x^d, and Q is a polynomial in y of degree c >= 2 satisfying d>c.
The germ f induces a tree map f on the valuative tree V_x. By results of Gignac-Ruggiero, all valuations of finite skewness have orbit converging to the eigenvaluation ord_x. Our goal is to describe the set K(f) of valuations (of infinite skewness) whose orbit does not converge to ord_x.
This set K(f) can be reinterpreted as the Julia set of a twisted polynomial map acting on the completion of the field of Puiseux series.
In our situation, we show that if K(f) does not contain critical curves, then it consists of curve semivaluations of uniformly bounded multiplicity.
This is part of a joint project with Romain Dujardin and Charles Favre. |
