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Chaos in uniform spaces

Let $Dom$ be a uniform space, and $f$ be a continuous function from $Dom$ to itself satisfying:

  1. For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural number $n$ and a member $x$ of $U$ such that $f^n(x)$ is a member of $V$.

  2. The periodic points of $f$ are dense in $Dom$.

Does it follow that $f$ satisfies (3)?

(3) There exists an entourage $U$ of $Dom$ such that for all members $x$ of $Dom$ and all neighborhoods $N$ of $x$, there exists a member $y$ of $N$ and a natural number $n$ such that $(f^n(x),f^n(y))$ is not a member of $U$.

According to on_intervals, the implication holds in metric spaces.

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