Are Periodic Orbit all Iterations?

Question

We have,

If $x$ is a periodic point of a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the period is $k$ i.e. $f^k(x)=x$ but $f^n(x)\not=x, \forall n: 0<n<k$.

(The statement up to this is slightly different in the text. This is my generalization but the main idea is same.)

Next the text says,

The periodic orbit of $x$ is $O(x)=\{x,f(x),f^1(x),f^2(x),\dots,f^{k-1}(x)\}$.

My question is, does it not have to be true that $f(x)=x$ to be periodic orbit? if $f^1(x)\not=x$ is it still in the periodic orbit?

Answer 1

I am guessing $f^k(x)$ means that you are iterating the function $f$ $k$ times. If that is the case, and $k=1$, then the function is of period one, actually the fundamental or the minimum period is one, then it is also 2,3,... periodic. Nothing would be wrong with the definition (saying that if $x=f(x)$ and you say is 1-periodic).

Next $O(x)$ refers precisely to the orbit, this is, the points in one fundamental period, that's why it says: $O(x)=\left\{x,f(x),f1(x),f^2(x),…,f^{k−1}(x)\right\}$, these are all the points within one period, and since, by definition, $x=f^k(x)$, then $f^k(x)$ is not included in the orbit.

Hope it helps.