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 (*edit*: in the Euclidean metric on $\mathbb{C}$). But the expanding constant can't literally be this lower bound, because it 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.