TITLE: “is topologically mixing” vs. “is topologically transitive” in the defition of chaos
QUESTION [9 upvotes]: 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?

REPLY [5 votes]: Another important example of a topologically transitive and not topologically mixing system (although it does not answer exactly your question) is the suspension flow over a topologically mixing system. 
Imagine that $f:X\to X $ is mixing. Then construct the suspension space
$X\times\mathbb{R}$, and the space $Y$ obtained after  quotient by the equivalence relation $(x,1)\sim (f(x),0)$. And consider the flow $\phi_t:(x,s)\in Y\mapsto (x,t+s)$. 
This dynamical system is topologically transitive, has dense orbits if it is the case for $f:X\to X$, but is not topologically mixing.
But of course, it is a flow, not a map.