| Abstract: |
| I wil consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. I will talk about two classes of such systems: attracting and parabolic. The latter being treated by means of the former.
I will provide fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits
ordered hy a natural geometric weighting.
These results will have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups.
Our approach is founded on spectral properties of complexified Ruelle--Perron--Frobenius operators and Tauberian theorems as used in classical problems of prime number theory. |
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