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.

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    $\begingroup$ Why do you say "it can't literally be that"? Sure, it's quite possible that a poorly chosen neighborhood will yield a lower bound that is $\le 1$. However, some better chosen neighborhood will yield a better lower bound, and as the neighborhood gets closer and closer to the Julia set $J$ itself, the lower bound on the neighborhood will approach $\min_{x \in J} |f'(x)|$. $\endgroup$ – Lee Mosher Jan 27 at 14:43
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    $\begingroup$ This reference might be helpful: MR2109468 (2005g:37084) Hruska, Suzanne Lynch Constructing an expanding metric for dynamical systems in one complex variable. (English summary) In particular, in Section 6 the author introduces and computes a "box-expanding" constant. Nonlinearity 18 (2005), no. 1, 81–100 $\endgroup$ – Margaret Friedland Jan 27 at 17:24
  • $\begingroup$ @LeeMosher Thanks for your comment. I may be missing something, but there are examples (including the rabbit) where there are points in the Julia set where $|f'(x)|<1$---when using the usual Euclidean metric on $\mathbb{C}$. That was the metric I meant to be referring to in my note. (I've edited to reflect this). If we instead take the "correct" metric in which the map is expanding, yes, your comment is I believe just the right thing. $\endgroup$ – Justin Lanier Jan 27 at 23:33

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