On mixing and dense periodicity on spaces with a free arc

We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval ( 0 , 1 ) (0,1) ). We provide a new proof of a result of M. Dirbák, L’. Snoha, V. Špitalský [Ergodic Theory Dynam. Systems, vol. 33 (2013), no. 6, pp. 1786–1812] stating that every continuous and transitive, but non-minimal map of a space with a free interval is relatively mixing, non-invertible, has positive topological entropy, and dense periodic points. The key simplification comes from short proofs of two facts. The first states that every weakly mixing map of a space with a free interval must be mixing and have positive entropy. The second states that a transitive but not minimal map of a space with a free interval has dense periodic points and is non-invertible.