It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is that all of the rational map's critical points be periodic. (See, e.g., Chapter 19 of Milnor's "Dynamics in one complex variable".)

Take as a specific example the Douady rabbit polynomial, $f(z) \approx z^2 -0.12+.074i$.

Expanding here means that there is a constant $k>1$ so that distances grow by at least a factor of $k$ under the map on a neighborhood of its Julia set in the associated Riemannian metric.

My question is: how can one compute a (relatively sharp) expanding constant $k$ for a given rational map? Say, concretely, for the rabbit polynomial? An answer or a relevant reference would be greatly appreciated!

Note: finding the expanding constant should have something to do with finding a lower bound on the derivative of the map in the neighborhood of the Julia set. But it can't literally be this, because this lower bound may not be bigger than 1. There should be some kind of uniformization to a hyperbolic metric happening, and some kind of effective Schwarz–Ahlfors–Pick theorem, but I'm not at all sure how to even begin such a calculation.


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