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Asymptotic counting in conformal dynamical systems. (English) Zbl 1482.37002

Memoirs of the American Mathematical Society 1327. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-6577-3/pbk; 978-1-4704-6632-9/ebook). v, 139 p. (2021).
The authors work in the general setting of conformal graph directed Markov systems over a countable alphabet. These are modeled by countable state symbolic subshifts of finite type. Their goal is to prove asymptotic counting results for multipliers and diameters corresponding to preimages or periodic orbits with a natural geometric weighting.
The general setting is as follows. Let \(\varphi_e: X \rightarrow X \) (with \(e\) in some finite or countable space \(E\)) be a family of conformal iterated functions on a compact subset \(X \in \mathbb{R}^d\) with \(d \geq 1\). For \(\xi \in X\) the authors define \(\varphi_\omega (\xi) = \varphi_{\omega_1} \circ \cdots \circ \varphi_{\omega_n} (\xi)\) where \(\omega = (\omega_1, \dots, \omega_n) \in E^n\) with \(n \geq 1\). The weights are defined as \(\lambda_\xi(\omega) = - \log |(\varphi_\omega)' (\xi)|\) and \(\Delta_\xi(\omega) = - \log \mathrm{diam} (\varphi_\omega (X))\).
A primary result is that, under very natural hypotheses, there are positive constants \(C_1\) and \(C_2\) and \(\delta \in (0, \infty)\) such that \[\lim_{T \to +\infty} \frac{\# \{ \omega: \lambda_\xi(\omega) \leq T\} }{e^{\delta T}} = C_1,\] and \[\lim_{T \to +\infty} \frac{\# \{ \omega: \Delta_\xi(\omega) \leq T\} }{e^{\delta T}} = C_2.\] The authors also provide explicit dynamical expressions for these constants. While these are the primary results, many others are presented and proved. For example, similar asymptotic results are developed under a requirement that the points \(\varphi_\omega(\xi)\) fall inside a prescribed ball in \(X\).
Prior work in [R. D. Mauldin and M. Urbanski, Trans. Am. Math. Soc. 351, No. 12, 4995–5025 (1999; Zbl 0940.28009)] provides a symbolic viewpoint and a framework to keep track the quantities the authors want to count using the concepts of attractive and parabolic countable alphabet conformal directed Markov systems.
The authors’ approach relies on the spectral properties of complexified Ruelle-Perron-Frobenius operators and associated Tauberian theorems. They note that there are natural analogues between their work and the classical approaches to the prime number theorem.
The main theme in this monograph is a description of a new and more general method to provide a unified approach to counting problems and statistical results such as central limit theorems. The authors’ aim is also to offer some applications and to suggest a variety of examples. Among these examples the authors discuss Apollonian circle packings, expanding and parabolic rational functions, and Kleinian groups.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37A50 Dynamical systems and their relations with probability theory and stochastic processes
37A44 Relations between ergodic theory and number theory
37B10 Symbolic dynamics
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37E25 Dynamical systems involving maps of trees and graphs
11N45 Asymptotic results on counting functions for algebraic and topological structures

Citations:

Zbl 0940.28009
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References:

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