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 $(0,1)$). We provide a new proof of a result of M. Dirb\'ak, \v{L}. Snoha, V. \v{S}pitalsk\'y [Ergodic Theory Dynam. Systems, vol. 33 (2013), no. 6, pp. 1786--1812] saying 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 says that every weakly mixing map of a space with a free interval must be mixing and have positive entropy. The second says that a transitive but not minimal map of a space with a free interval has dense periodic points and is non-invertible.