This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours.

Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits"

as the definition of chaos, and this paper shows that its (the paper's) definition of chaos

is equivalent to "is topologically transitive and has dense periodic orbits".

Clearly, every topologically mixing map is topologically transitive, and there are topologically transitive maps that are not topologically mixing (such as an irrational rotation of the circle).

Is there a map that is topologically transitive *and* has dense periodic orbits

but is not topologically mixing? $\:$ If yes, can its space be metrizble?