# Asymptotic Counting in Conformal Dynamical Systems

## TL;DR

This paper develops asymptotic counting results and central limit theorems for conformal graph directed Markov systems, with applications to various fractal and dynamical systems, using spectral analysis of Ruelle operators.

## Contribution

It introduces a new spectral approach to asymptotic counting in conformal dynamical systems, covering both attracting and parabolic cases.

## Key findings

- Asymptotic formulas for multipliers and diameters of preimages and periodic orbits.
- Central Limit Theorems describing distribution of weights.
- Applications to systems like Apollonian packings, Farey maps, and Fuchsian groups.

## Abstract

In this article 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 hy a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These have direct applications to a 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. A fairly complete collection of asymptotic counting results for them is presented. 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.