It is known that some real Cantor Julia sets appear as spectra
of Markov operators
on self-similar graphs. Examples of such graphs are:
- Sierpinski graph (Grabner-Woess, 97);
- Schreier graphs of some self-similar groups (Bartholdi-Grigorchuk, 00);
- Schreier graphs of the Hanoi towers group on three pegs
(Grigorchuk-Sunic, 08).

Moreover, spectral measures of the corresponding Markov operators are
closely related
to Brolin-Lyubich measures on the Julia sets. This motivates studying
the moments of Brolin-Lyubich measures.
In this talk following the approach of Grabner-Woess I will obtain a
description of the asymptotic behavior
of these moments for quadratic maps $z^2+c$ with $c<-2$, which includes all
of the above mentioned examples.