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?


The answers to both questions are: yes. There are many examples. The simplest is maybe this: Let $T(x)=1-|1-2x|$ be the tent map. Take the product space $[0,1]\times\{0,1\}$ and the map $F(x,y)=(T(x),1-y)$, that is, take the Cartesian product of the unit interval with two-element set and the map which is a product of the tent map and a cyclic permutation $0\mapsto 1$, $1\mapsto 0$. This is a topologically transitive system with dense set of periodic points on a compact metric space (the verification of this is easy). It is not topologically mixing since its second iterate $F^2$ is not transitive. Another example: define $$ f(x)=\begin{cases}1/2+2x,& 0\le x \le 1/4,\\ 3/2-2x,& 1/4<x\le 1/2\\ 1-x,&1/2<x \le 1.\end{cases} $$ This map is transitive and has dense set of periodic points, but the second iterate is not transitive (it is known that for the continuous interval map topological mixing is equivalent to transitivity of the second iterate).

  • $\begingroup$ The second example is so interesting! $\endgroup$ – Pengfei Sep 28 '13 at 20:41

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.

  • 1
    $\begingroup$ Nice example. And I think that, by taking the time-$\epsilon$ map of the suspension flow for $\epsilon$ small, we can also obtain an example of transitivity and dense orbits without mixing for a map rather than a flow. $\endgroup$ – Matthew Kvalheim Jul 19 at 5:53

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