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Entropy for $k$-trees defined by $k$ transition matrices

Andressa Paola Cordeiro, Alexandre Tavares Baraviera, Alex Jenaro Becker

TL;DR

This work develops a unified framework to analyze the entropy of Markov tree-shifts defined by $k$ transition matrices. By introducing the dynamical system $f(x)=(A_1x)*(A_2x)*\cdots*(A_kx)$ and proving $p(n)=\|f^n(\mathbf{1})\|$, the authors provide a constructive method to compute both $h_{BC}$ and $h_{PS}$ and compare these entropies under conjugacy and block-tranformation. They derive general upper bounds for $h_{PS}$, investigate its behavior across many binary examples, and establish criteria for the existence of invariant measures, while extending topological properties such as irreducibility and mixing to tree-shifts. The paper also situates these results within the open problems proposed by Ban and Chang, offering explicit classifications and numerical estimates that illuminate when one entropy dominates or when positive entropy implies stronger dynamical features. Overall, the approach links combinatorial complexity with nonlinear dynamical iterations, yielding practical algorithms for entropy computation and deeper insight into the ergodic structure of tree-shifts.

Abstract

We study Markov tree-shifts given by $k$ transition matrices, one for each of its $k$ directions. We provide a method to characterize the complexity function for these tree-shifts, used to calculate the tree entropies defined by Ban and Chang arXiv:1509.08325 and Petersen and Salama arXiv:1712.02251. Moreover, we compare these definitions of entropy in order to determine some of their properties. The characterization of the complexity function provided is used to calculate the entropy of some examples. The question of existence of a specific type of invariant measures for such tree-shifts is addressed. Finally, we analyse some topological properties introduced by Ban and Chang arXiv:1509.01355 for the purpose of answering two of the questions raised by these authors.

Entropy for $k$-trees defined by $k$ transition matrices

TL;DR

This work develops a unified framework to analyze the entropy of Markov tree-shifts defined by transition matrices. By introducing the dynamical system and proving , the authors provide a constructive method to compute both and and compare these entropies under conjugacy and block-tranformation. They derive general upper bounds for , investigate its behavior across many binary examples, and establish criteria for the existence of invariant measures, while extending topological properties such as irreducibility and mixing to tree-shifts. The paper also situates these results within the open problems proposed by Ban and Chang, offering explicit classifications and numerical estimates that illuminate when one entropy dominates or when positive entropy implies stronger dynamical features. Overall, the approach links combinatorial complexity with nonlinear dynamical iterations, yielding practical algorithms for entropy computation and deeper insight into the ergodic structure of tree-shifts.

Abstract

We study Markov tree-shifts given by transition matrices, one for each of its directions. We provide a method to characterize the complexity function for these tree-shifts, used to calculate the tree entropies defined by Ban and Chang arXiv:1509.08325 and Petersen and Salama arXiv:1712.02251. Moreover, we compare these definitions of entropy in order to determine some of their properties. The characterization of the complexity function provided is used to calculate the entropy of some examples. The question of existence of a specific type of invariant measures for such tree-shifts is addressed. Finally, we analyse some topological properties introduced by Ban and Chang arXiv:1509.01355 for the purpose of answering two of the questions raised by these authors.
Paper Structure (20 sections, 19 theorems, 101 equations, 1 table)

This paper contains 20 sections, 19 theorems, 101 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation and main definitions
  3. Characterization of the complexity function
  4. Comparing the entropies $h_{BC}$ and $h_{PS}$ for tree-shifts
  5. Upper and lower bounds for the entropy $h_{PS}$ of Markov tree-shifts
  6. The entropy of all binary tree-shifts over the alphabet $\{0,1\}$
  7. Case 1.
  8. Case 2.
  9. Case 3.
  10. Case 4.
  11. Case 5.
  12. Case 6.
  13. Case 7.
  14. Case 8.
  15. Case 9.
  16. ...and 5 more sections

Key Result

Proposition 2.1

Any tree-shift of finite type consisting of $k$-trees, up to an alphabet change by a suitable $m$-block map, is a Markov tree-shift whose allowed blocks can be characterized by matrices $A_1, \dots, A_k$, all with the same order.

Theorems & Definitions (50)

  • Proposition 2.1
  • Theorem 3.1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 4.1
  • proof
  • Remark 3
  • proof
  • Remark 4
  • ...and 40 more