Iterates converging to a continuous map

Question

I have no doubt that the following observation is quite well known. Let $\varphi:[0,1]\to [0,1]$ be a continuous map. Assume that the iterates $\varphi^n$ converge pointwise to some continuous map $\varphi_\infty$. Then the convergence is in fact uniform. However, I was unable to locate a reference. Does anybody know where I can find it written? Thanks in advance.

Answer 1

I don't know a reference but maybe the following proof is shorter than yours.

By continuity, $\varphi\circ\varphi_\infty=\varphi_\infty$. Hence $\varphi$ is identity on the set $I:=\varphi_\infty([0,1])$. Hence $I$ is the set of fixed points of $\varphi$. And it is compact and connected. Then there are two cases: either $I$ is a nontrivial interval $[a,b]$ or it is a single point $\{a\}$.

In the first case, we have $f(x)<x$ for all $x>b$ and $f(x)>x$ for all $x<a$, otherwise there is a fixed point outside $I$. There is an $\varepsilon>0$ such that $f(x)>a$ for all $x\in(b,b+\varepsilon)$ and $f(x)<b$ for all $x\in(a-\varepsilon,a)$. Let $U=(a-\varepsilon,b+\varepsilon)$. Then $\varphi(U)\subset U$ and for every $x\in U$ we have $$ |\varphi^n(x)-x|<\delta(\varepsilon) $$ where $\delta(\varepsilon)\to $ as $\varepsilon\to 0$ ($\delta(\varepsilon)$ is the maximum of the diameters of the sets $\varphi([a-\varepsilon,a])$ and $\varphi([b,b+\varepsilon])$) For every $x\in[0,1]$, the iterations $\varphi^n(x)$ eventually get into $U$, hence $$ \bigcup_n \varphi^{-n}(U) = [0,1] . $$ By compactness, this implies that $[0,1]$ is covered by finitely many sets $f^{-n}(U)$ and hence there exist $n$ such that $\varphi^n([0,1])\subset U$. After this $n$, all iterations stay within the distance $\delta(\varepsilon)$ from the limit. Since $\varepsilon$ can be arbitrarily small, this implies uniform convergence.

In the second case (when $I$ is a single point $a$), similarly observe that $f(x)<x$ for $x>a$ and $f(x)>x$ for $x<a$, and the same holds for $\varphi^2$, $\varphi^3$, etc. This implies that if e.g. $x>a$ and $f(x)<a$ then all iterations $\varphi^n(x)$ lie between $x$ and $f(x)$. This easily implies that there is an arbitrarily small neighborhood $U$ of $a$ such that $f(U)\subset U$. Then uniform convergence follows similarly to the first case.

Answer 2

This result seems to be a consequence of Dini's theorem (as noted by nonlinearism):

If $\varphi^n$ converges to a continuous function $\varphi_\infty$, we have $\varphi_\infty = 0$. Hence, $\varphi < 1$ on $[0,1]$. By compactness, we have $\varphi < p$ for some $p < 1$. Therefore, the convergence is monotonic and we can apply Dini's theorem.

This should work for all $\varphi : K \to [-1,1]$, where $K$ is a compact metric space (for $\varphi < 0$, one has to adopt the proof in an obvious manner).