In relation to this question: Relation between volume entropy and Hausdorff dim of limit set?

Given a metric space $Z$ and a hyperbolic approximation $X := hyp_{r_0}(Z)$ (as defined for example here).

I noticed the following correspondence (for $x_0 \in X$):

$$ \lim_{r \to \infty} \frac{1}{r} \log_{1/r_0}(vol(B_r(x_0))) = \dim_H(Z).$$

Is this generally true? Does this follow from the answer in the linked question?


Found the answer: This follows directly from Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov by Michel Coornaert (https://projecteuclid.org/download/pdf_1/euclid.pjm/1102634263):

If we define $$e_a(\Gamma) = \limsup_{n \to \infty} \frac{\log \#\{g \in \Gamma : l(g) \leq n\}}{n \log a}$$

Then we have: $\dim_H(\partial \Gamma) = e_a(\Gamma)$ where $a = \frac{1}{r_0}$.


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