On entropies of block-gluing subshifts

TL;DR

This paper investigates the topological entropies of c-block gluing subshifts, showing the set of all such entropies is dense, but specific subsets like R_1 and R_2 have isolated points, indicating complex entropy structures.

Contribution

It introduces the sets of entropies for c-block gluing subshifts and proves their density, revealing intricate properties of these dynamical systems.

Findings

The set R of all c-block gluing entropies is dense.

The sets R_1 and R_2 are not dense and contain isolated points.

Conjecture that the same properties hold for all c.

Abstract

A subshift $X$ is called $c$-block gluing if for any integer $n\geq c$ and any two blocks $u$ and $v$ from the language of $X$ there exists an element of $X$ which has occurrences of $u$ and $v$ at distance $n$. In this note we study the topological entropies of $c$-block gluing binary one-dimensional subshifts. We define the set $R_c$ to be the set of entropies of all $c$-block-gluing subshifts, and $R=\cup_{c\in \mathbb{N}} R_c$. We show that the set $R$ is dense, while $R_1$ and $R_2$ are not; in particular, they have isolated points. We conjecture that the same holds for any $c$.