Iterations of rational functions: which hyperbolic components contain polynomials?

1994/04/09 by Przytycki, Feliks
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/9404237

Abstract

Let H^d be the set of all rational maps of degree d≥ 2 on the Riemann sphere which are expanding on Julia set. We prove that if f∈ 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 ≥ 2 of critical points in the basin. For such a map f∈ H^d with k

Related

• Julia sets as buried Julia components
• Hyperbolic components in spaces of polynomial maps
• Singular perturbations with multiple poles of the simple polynomials
• Boundaries of capture hyperbolic components
• On the cycles of components of disconnected Julia sets