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

(1)<br>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.
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(2)<br>The periodic points of f are dense in Dom.

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Does if follow that f satisfies (3)?

(3)<br>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.

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According to pb.math.univ.gda.pl/chaos/pdf/on_intervals.pdf, the implication holds in metric spaces.