Trees in positive entropy subshifts

TL;DR

This paper proves that positive entropy subshifts necessarily contain infinite binary trees with branches that split synchronously at times forming a set with positive lower density.

Contribution

It provides a simple proof establishing the existence of structured infinite trees within positive entropy subshifts, highlighting their combinatorial richness.

Findings

Positive entropy subshifts contain infinite binary trees with synchronized branching.

Branching times in these trees have positive lower asymptotic density.

The proof simplifies understanding of the structure within positive entropy subshifts.

Abstract

I give a simple proof for the fact that positive entropy subshifts contain infinite binary trees where branching happens synchronously in each branch, and that the branching times form a set with positive lower asymptotic density.