Abstract
We look at the maximal entropy measure (MME) of the boundaries of connected components of the Fatou set of a rational map of degree $≥ 2$. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness.
Highlights
Let R : C → C be a rational map of degree d ≥ 2, defined on the Riemann sphere C, F its Fatou set, J its Julia set, and λ the unique maximal entropy measure (MME) on J (i.e., hλ(R) = log d)
In the jointly written appendix of [5], the authors showed that the backward random iteration method works for drawing the Julia set of any rational map
We give conditions under which we can show that, if O is another component of F, i.e., a bounded component of F, λ(∂O) = 0. We demonstrate this for a class of polynomials whose Fatou sets have an infinite number of components
Summary
Let R be a rational map of degree ≥ 2, and assume F has infinitely many connected components. 3. Other maps with a completely invariant Fatou component. Extending our scope a bit, let us assume that R (of degree d ≥ 2) has the property that F has a completely invariant connected component O#, i.e., (3.1)
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