Let f be a continuous function of a metric space, M, to itself with a dense orbit and a fixed point.(adsbygoogle = window.adsbygoogle || []).push({});

I.e. there exists z such that the set {f^{(n)}(z)} for all n ∊ N (where f^{(n)}is the nth iterate of f) is dense in M, and there exists p such that f(p) = p.

Does this imply that f spreads?

I.e. does there exists δ > 0 such that for x ≠ y there is an n_{0}(depending on x and y) with |f^{(n0)}(x) - f^{(n0)}(y)| > δ.

I can prove this for M = R or S^{1}, but I don't know in general, or even in the case where M = the orbit and p.

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# I Dense orbit and fixed point question

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