# Diophantine approximation in the Julia set

Let $f : \mathbb{CP}^1 \to \mathbb{CP}^1$ be a rational map of degree $q > 1$; or just a quadratic binomial $z^2 + c$, if one prefers. The Julia set $J_f$ is the closure of the repelling periodic points of $f$, which are equidistributed in the maximum entropy measure $\mu_f$ (of Brolin-Lyubich); the other (non-repelling) periodic points are finitely many. So it makes sense to ask how closely may a point $x \in J_f$ be approached by a periodic point $y$.

For a measure of complexity of $y$ (in terms of which to assess the quality of the approximation), let me consider not the period but the size of the Galois orbit: Letting $F/\mathbb{Q}$ the finitely generated field extension adjoining the coefficients of $f$, consider $d(y) := [F(y):F]$. The case $f = z^2$ suggests this quantity to be somewhat better behaved in this question; and this also takes care of the preperiodic (finite orbit) points. If one prefers to consider $d(y) = n$ in the following, for $y$ a periodic point of period $n$, I am equally happy with that. (When considering a preperiodic point of preperiod $m$ and period $n$, it will be $nq^m$ that roughly corresponds to my $d(y)$. To compare these, in the $f = z^2$ example it is easily seen that $\sqrt{d(y)} \ll nq^m \ll d(y)$ for all but a negligible fraction of $y \in \mu_{\infty}$. So it might not make too much of a difference which measure of complexity is chosen.)

In the case $f = z^2$ we may reformulate Dirichlet's and Khinchin's theorems on rational approximations to real numbers as follows. Every non-preperiodic $x \in S^1$ admits infinitely many periodic approximants $y$ with $|x-y| < d(y)^{-2}$, and for all $K > 2$, Lebesgue almost all $x \in S^1$ have $|x-y| \gg_x d(y)^{-K}$ for all (pre)periodic points $y$.

Extending this to an arbitrary $f$, let me say $x \in J_f$ is Diophantine if there is a $K < \infty$ such that $|x-y| \gg_x d(y)^{-K}$ for all periodic points $y$. And let me say $x$ is approachable if there is a $k > 0$ such that $|x-y| < d(y)^{-k}$ has infinitely many solutions in periodic points $y$.

Question: (i) Are $\mu_f$-almost all points Diophantine? (With a uniform exponent $K$?) (ii) Is every point of $J_f$ approachable?