Arcwise connectedness of the set of ergodic measures of hereditary   shifts
  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. 
Jakub Konieczny, Michal Kupsa and Dominik Kwietniak
math.DS