Spivak: $f$ cont. on $[0,1]$, $0\leq f(x)\leq 1$ for all $x\in [0,1]$. Examples of $f$ such that the seq $x,f(x),f(f(x)),...$ does not converge? Let $f$ be continuous on $[0,1]$ and $0\leq f(x)\leq 1$ for all $x\in [0,1]$.
It can be shown that $f$ has a fixed point in $[0,1]$.
Now consider the sequence
$$x,f(x),f(f(x)), f(f(f(x))),...\tag{1}$$
What are some examples of a function $f$ such that this sequence does not converge for at least one $x$ in $[0,1]$?
I found one example

Are there others?
Note that by this result, if the sequence does converge, then it is to the fixed point.
Context of why I am asking this.
The following is a problem from Ch. 22 of Spivak's Calculus

21.(a) Suppose that $f$ is continuous on $[0,1]$ and that $0\leq f(x)\leq 1$ for all $x\in [0,1]$. Problem 7-11 shows that $f$ has a
fixed point. If $f$ is increasing, a much stronger statement can be
made: For any $x$ in $[0,1]$, the sequence $x, f(x), f(f(x)),...$ has
a limit (which is necessarily a fixed point, by Problem 20). Prove
this assertion.

This question asks how we can visualize the convergence of the sequence $(1)$. I am asking how can visualize non-convergence.
EDIT:
After reading Sergei Nikolaev's answer, here is another example based on that answer.

As he mentions, we can come up with an infinite number of examples of this sort.
 A: Example 1. Fix $a \in (0,1) \backslash \{ \frac12 \}$. Consider any continious $f(x)$ such that $f(a) = 1- a$ and $f(1-a) = a$.
This idea may be generalized.
Example 2. Fix $N \ge 1$. Consider $a_1, a_2, \ldots, a_N \in (0,1)$, $a_i \ne a_j$. Consider any continious $f(x)$ such that $f(a_i) = a_{i+1}$, $1 \le i \le N-1$ and $f(a_N) = a_1$.
These are examples, when $x, f(x), f(f(x)), \ldots$ does not converge for some $x$.
A: Here is a recipe for synthetic examples:
For any distinct numbers $0= x_0<\ldots<x_n=1$, define a function such that
$x_k=f(x_{k-1})$ for $1\leq k\leq n$ and $f(x_n)=f(x_0)$. Then define $f$ on $x_{k-1}<t<x_k$ linearly. That will give you a sequence such that the orbit of $x_0$, $\{f^{\circ n}(x_0):n\in\mathbb{N}\}$ does not converge. In fact the orbit moves cyclicly from $x_0$ to $x_1$ ... to $x_n$ and back to $x_0$. (Here $f^{\circ n}$ means composing $f$ with itself $n$ times).
For a more interesting example that has been studied a lot in Mathematics and has a very rich pedigree, consider the quadratic maps $$f_r(x)=rx(1-x)$$
where $0<r\leq 4$ and $0\leq x\leq 1$. All kinds of interesting phenomemna is observed as the parameter $r$ varies from $0+$ to $4$.
