Arcwise connectedness of the set of ergodic measures of hereditary shifts

TL;DR

This paper proves that the set of ergodic measures in hereditary shifts with a safe symbol is arcwise connected under the d-bar metric, with the entropy function covering all values between zero and the topological entropy.

Contribution

It establishes the arcwise connectedness of ergodic measures in hereditary shifts and confirms the entropy function's range, addressing a conjecture of A. Katok.

Findings

Ergodic measures form an arcwise connected set under the d-bar metric.

The entropy function takes all values between zero and the topological entropy.

The results apply to all hereditary shifts with a safe symbol.

Abstract

We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the $d$-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A.~Katok.