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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?

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