Iterations of rational functions: which hyperbolic components contain polynomials?

Abstract

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the one-sided shift space of $d$ symbols. We give exotic examples of maps of an arbitrary degree $d$ with a non-simply connected, completely invariant basin of attraction and arbitrary number $k \ge 2$ of critical points in the basin. For such a map $f\in H^d$ with $k<d$ there is no polynomial in $H(f)$. Finally we describe a computer experiment joining an exotic example to a Newton's method (for a polynomial) rational function with a 1-parameter family of rational maps.