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On the topological pressure of axial product on trees

Jung-Chao Ban, Yu-Liang Wu

TL;DR

The paper addresses the problem of understanding the topological pressure for isotropic axial products of Markov subshifts on $d$-trees. It develops a pattern-distribution framework and a finite-k optimization $P^{(k)}(d,E)$, proving that the limiting pressure $P^{(\infty)}(d,E)$ is continuous and increasing in $d$, with asymptotic limits $\log \rho(E)$ and $\log r_E$ as $d\to1^+$ and $d\to\infty$, respectively. The authors transform the combinatorial problem using Stirling's approximation and KL-divergence, derive an explicit entropy-maximizing form for the optimizer, and establish convergence of $P^{(k)}(d,E)$ to $P^{(\infty)}(d,E)$ along with a rigorous monotonicity proof. They also provide numerical experiments on golden-mean tree-shifts to verify the theoretical results and demonstrate applicability to broader shift spaces. The work extends limiting pressure concepts to tree-shifts and offers a versatile framework for analyzing high-dimensional axial products through a variational, pattern-distribution lens.

Abstract

This article investigates the topological pressure of isotropic axial products of Markov subshifts on the $d$-tree. We show that the quantity increases with dimension $d$. To achieve this, we introduce the pattern distribution vectors and the associated transition matrices and partially transplant the large deviation theory to tree-shifts. Additionally, we apply our main result to a broader class of shift spaces, accompanied by numerical experiments for verification.

On the topological pressure of axial product on trees

TL;DR

The paper addresses the problem of understanding the topological pressure for isotropic axial products of Markov subshifts on -trees. It develops a pattern-distribution framework and a finite-k optimization , proving that the limiting pressure is continuous and increasing in , with asymptotic limits and as and , respectively. The authors transform the combinatorial problem using Stirling's approximation and KL-divergence, derive an explicit entropy-maximizing form for the optimizer, and establish convergence of to along with a rigorous monotonicity proof. They also provide numerical experiments on golden-mean tree-shifts to verify the theoretical results and demonstrate applicability to broader shift spaces. The work extends limiting pressure concepts to tree-shifts and offers a versatile framework for analyzing high-dimensional axial products through a variational, pattern-distribution lens.

Abstract

This article investigates the topological pressure of isotropic axial products of Markov subshifts on the -tree. We show that the quantity increases with dimension . To achieve this, we introduce the pattern distribution vectors and the associated transition matrices and partially transplant the large deviation theory to tree-shifts. Additionally, we apply our main result to a broader class of shift spaces, accompanied by numerical experiments for verification.
Paper Structure (12 sections, 10 theorems, 109 equations, 2 figures)

This paper contains 12 sections, 10 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov tree-shifts and topological pressure
  4. Pattern distribution
  5. Topological pressure and pattern distribution
  6. Step 1
  7. Step 2
  8. Step 3
  9. Step 4
  10. Convergence of $P^{(k)}(d,E)$
  11. Proof of Theorem \ref{['thm:pressure_monotonicity']}
  12. Experiments

Key Result

Theorem 1

Suppose that the interaction matrix $E$ satisfies eq:assumption. Then, the following assertions hold.

Figures (2)

  • Figure 1: Topological entropy of golden-mean tree-shifts
  • Figure 2: Topological pressure of golden-mean tree-shifts

Theorems & Definitions (22)

  • Theorem 1
  • Corollary 2
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 12 more