Improved Bounds For Some Asymptotic Formulas for Counting Words in Shift Spaces

TL;DR

This paper refines asymptotic formulas for counting words in shift spaces within conformal dynamical systems, removing previous restrictions and providing sharper bounds using recent Tauberian theorems.

Contribution

It extends existing results by lifting the D-generic and conformality conditions, offering improved bounds and insights into word and length asymptotics in shift spaces.

Findings

Asymptotic bounds for word counting are sharpened.

The length of words also exhibits related asymptotic behavior.

The results apply to a broader class of conformal dynamical systems.

Abstract

This paper studies a version of the counting problem in dynamical systems that is of interest, especially in conformal dynamical systems where the functions of the systems are angle preserving. Recently, M. Pollicott and M. Urba\'{n}ski published a result in this context for D-generic systems where the complex transfer operator behaves nicely on the critical line of the Poincar\'{e} series. Their result contains an asymptotic formula for the Apollonian circle packing. We lift the D-generic condition and conformality of the functions system in this paper to see how their asymptotic formula changes. We use some recent Tauberian theorem to show that the formula gets a form whose limit infimum and limit supremum bounds can be obtained in the sharpest sense. Further, we observed an asymptotic of length closely related to this counting problem. In fact, not only the number of words is subject to some formula, but also their length as well.