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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.

Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ 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 $Dom$.

$\;\;$3.$\:\:$There exists an entourage $E$ of $Dom$ such that for all members $x$ of
$\quad \;\;$ $Dom$ and all neighborhoods $U$ of $x$, there exists a member $y$ of $U$ and
$\quad \;\;$ a natural number $n$ such that $\:\langle \hspace{.05 in}f^n(x)\hspace{.02 in},\hspace{.03 in}f^n(\hspace{.03 in}y)\rangle\:$ is not a member of $E$.

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

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