Hereditary subshifts whose simplex of invariant measures is Poulsen

TL;DR

This paper identifies conditions under which the set of invariant measures for hereditary systems forms a Poulsen simplex, highlighting positive entropy -free systems as a key example, and provides a counterexample.

Contribution

It establishes a sufficient condition for hereditary systems' invariant measure simplices to be Poulsen and explores specific cases including -free systems.

Findings

The simplex is Poulsen for positive entropy -free systems.

A hereditary system with positive entropy can have a non-Poulsen simplex.

Provides a counterexample illustrating the limits of the main result.

Abstract

We give a sufficient condition for the simplex of invariant measures for a hereditary system to be Poulsen. In particular, we show that this simplex is Poulsen in case of positive entropy $\mathscr{B}$-free systems. We also give an example of a positive entropy hereditary system whose simplex of invariant measures is not Poulsen.