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Let be $M$ a topological space, and $f:M\to M$ a danymical system, i.e, a continuous map between from $M$ to $M$.

We say that a dynamical system, $f:M\to M$ is topologically transitive when, exists $x\in M$ such that, $Orb(x)=\{x,f(x),\ldots, f^n(x),\ldots\}$ is dense in $M.$

There is a problem in the book of Brin Stuck, An introduction to Dynamical Systems, that makes following question: Is the product of two topologically transitive (minimal, topologically mixing) systems topologically transitive (minimal, topologically mixing)?

I already know that for minimal systems the answer is no, And as for mixing systems the answer is yes.

But I have no intuition for the case of topologically transitive systems, so my question is: Is the product of two topologically transitive, topologically transitive?

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1 Answer 1

up vote 1 down vote accepted

Let $M=\{0,1\}$ with the discrete topology, and let $f:M\to M:x\mapsto 1-x$. Clearly $\langle M,f\rangle$ is transitive. Let $$F=f\times f:M\times M\to M\times M:\langle x,y\rangle\mapsto\langle f(x),f(y)\rangle\;.$$ Then for each $p\in M\times M$, $|\operatorname{Orb}(p)|=2$, so $\operatorname{Orb}(p)$ is not dense in the $4$-point discrete space $M\times M$.

(In fact $M\times M$ is the union of the two disjoint $F$-orbits $\{\langle 0,0\rangle,\langle 1,1\rangle\}$ and $\{\langle 0,1\rangle,\langle 1,0\rangle\}$.)

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Very elegant. Cool! – user27456 Apr 30 '12 at 19:01

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