Pincement de polynômes

Abstract

Let $f_0:\mathbb{C}\to \mathbb{C}$ be a semi-hyperbolic polynomial in the sense of Carleson-Jones-Yoccoz with an attracting point. The goal of this paper is to show that one can define a semi-hyperbolic deformation $(f_t{)}_{t\ge 0}$ such that the attracting cycle becomes parabolic for the limit polynomial $f_{\infty }$ and that $f_0$ and $f_{\infty }$ are semi-conjugate. This deformation is defined by pinching curves in appropriate quotient spaces.