Institute for Mathematical Sciences
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
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