Arcwise connectedness of the set of ergodic measures of hereditary shifts

摘要

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