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I have a dynamical system $(X, f)$ where $f:X \rightarrow X$ and $X$ is a complete metric space. There exists an $x_0 \in X$ such that $f^k(x_0) \rightarrow x_0$ as $k\rightarrow \infty$, and all the $f^k(x_0)$ for $k > 0$ are distinct points in $X$.

Does anyone know how to classify such a dynamical system? From what I can tell, we cannot call such a system topologically ergodic, but it also seems wrong to say that the map has an infinite periodic orbit. Can anyone point me to ideas or literature that deal with such problems?

Edit: in this case $f$ is a homeomorphism, and is a contraction mapping on the interval $[0, 1)$. When $x_0 = 1$, $f(x_0) = 0$, and then $f^k(x_0) \rightarrow 1$ as $k\rightarrow \infty$.

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  • $\begingroup$ Is $f$ continuous (included in "dynamical system?)? Do you have a concrete example for such a situation or something close to it? $\endgroup$ May 10, 2020 at 19:34
  • $\begingroup$ @LutzLehmann yes $f$ is homeomorphism. I will the question to give an example that well demonstrates the situation. $\endgroup$
    – user918212
    May 10, 2020 at 20:06
  • $\begingroup$ Then that situation is impossible. If ever $f^k(x_0)$ comes close enough to $x_0$, then by continuity $f^{k+1}(x_0)$ will be close to $x_1$. You could have $x_0$ as one limit point of the orbit, but not as the limit. $\endgroup$ May 10, 2020 at 21:38

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Let $X$ be a Hausdorff space and $f\colon X\to X$ a continuous transformation. Suppose that $x_{0}\in X$ is a point such that $f^{k}(x_{0})\to x_{0}$. Then $x_{0}$ is a fixed point of $f$. Indeed, by continuity of $f$ we have $f(f^{k}(x_{0}))\to f(x_{0})$. On the other hand, we have $f(f^{k}(x_{0}))=f^{k+1}(x_{0})\to x_{0}$ by assumption. Since limits in Hausdorff spaces are unique, we conclude that $f(x_{0})=x_{0}$.

Note: Metric spaces are Hausdorff, so this applies to your situation.

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