Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Concatenations in subshifts defined by linear orders and poles of the Artin-Mazur zeta function

Chenxi Wu

TL;DR

The paper develops a symbolic-dynamics framework for concatenating dominant and admissible words on tunable graphs to realize persistence phenomena in entropy and poles of the Artin-Mazur zeta function. By introducing a twisted lexicographic order, admissibility criteria, and tuning (including unique maximal tuning), it proves that nonrenormalizable words can be extended to dominant forms and that the spectral data of the associated Markov decompositions can be controlled via prefix-suffix structure. A key construction is the polynomial $F_w(lambda)$ whose roots off the unit circle correspond to off-unit-circle eigenvalues, enabling Rouche-type estimates that link entropy to the location of zeta-poles. The results generalize persistence phenomena from interval maps to families of maps on finite trees, recovering known theorems in previous works and offering new tree-map applications and open questions about relaxing tunability assumptions toward broader Mandelbrot-set veins.

Abstract

For certain pairs of unimodal maps on the interval with periodic critical orbits, it is known that one can combine them to create another map whose entropy is close to one while the poles of Artin-Mazur $ζ$ function outside the unit circles can be made close to the other. We provided a formulation and proof of this result in symbolic dynamics setting, which allow us to generalize this fact to certain families of maps on finite trees.

Concatenations in subshifts defined by linear orders and poles of the Artin-Mazur zeta function

TL;DR

The paper develops a symbolic-dynamics framework for concatenating dominant and admissible words on tunable graphs to realize persistence phenomena in entropy and poles of the Artin-Mazur zeta function. By introducing a twisted lexicographic order, admissibility criteria, and tuning (including unique maximal tuning), it proves that nonrenormalizable words can be extended to dominant forms and that the spectral data of the associated Markov decompositions can be controlled via prefix-suffix structure. A key construction is the polynomial whose roots off the unit circle correspond to off-unit-circle eigenvalues, enabling Rouche-type estimates that link entropy to the location of zeta-poles. The results generalize persistence phenomena from interval maps to families of maps on finite trees, recovering known theorems in previous works and offering new tree-map applications and open questions about relaxing tunability assumptions toward broader Mandelbrot-set veins.

Abstract

For certain pairs of unimodal maps on the interval with periodic critical orbits, it is known that one can combine them to create another map whose entropy is close to one while the poles of Artin-Mazur function outside the unit circles can be made close to the other. We provided a formulation and proof of this result in symbolic dynamics setting, which allow us to generalize this fact to certain families of maps on finite trees.
Paper Structure (7 sections, 13 theorems, 15 equations, 3 figures)

This paper contains 7 sections, 13 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Concatenation of Dominant and Admissible Words
  4. Tuning and Renormalizability
  5. Construction of Dominant Words
  6. Topological entropy and Poles of the Artin-Mazur zeta function
  7. Applications to Dynamics and further questions

Key Result

Theorem 1.1

bray2021shapelindsey2021master Let $\mathcal{F}$ be a family of maps, which can be: Every map $f\in F$ admits a finite Markov decomposition. Let $M_f$ be its incidence matrix, $E_f$ the set of eigenvalues of $M_f$, and $h_f$, which is the $\log$ of the leading eigenvalue of $M_f$, the topological entropy. If $f_1, f_2, f_3\in \mathcal{F}$, $h_{f_1}<h_{f_2}<h_{f_3}$, then for any $\

Figures (3)

  • Figure 1: The intersection of Thurston's master teapot with the unit cylinder, plotting the half where the imaginary part is non negative.
  • Figure 2: The tree in Example \ref{['tree']}
  • Figure 3: The teapot for Example \ref{['tree']}, with the unit cylinder, plotting the half where the imaginary part is non negative.

Theorems & Definitions (44)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 34 more