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Asymptotic Counting in Conformal Dynamical Systems
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Asymptotic Counting in Conformal Dynamical Systems
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eBook
Asymptotic Counting in Conformal Dynamical Systems
2021
Overview
In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic
subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the
former.
We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic
orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features
of the distribution of their weights.
These results 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, and many more. This gives a unified approach which both recovers known
results and proves new results.
Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius
operators and Tauberian theorems as used in classical problems of prime number theory.
