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Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.

Let $S : X → X$ and $T : Y → Y$ be dynamical systems. Then the map $S × T$ is defined by:

$S × T : X × Y → X × Y$, $(S × T)(x, y) = (Sx, Ty)$.

We know that if $S$ and $T$ are topologically transitive and at least one of them is mixing then $S × T$ is topologically transitive (https://www.merry.io/dynamical-systems-lecture-notes/2016/10/3/the-shift-map).

My question is: Consider an infinite set of mixing maps $S_{i}:X_{i}→X_{i}$ along a topologically transitive map $f$.

Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive.

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