Periodic points of the dynamical system $(K,\varphi)$ and Periodic Maximal Ideals in its associated Koopman system $(C(K),T_{\varphi})$

Question

At the moment I am reading Operator Theoretic Aspects of Ergodic Theory by Eisner et al. and trying to understand the concept of periodicity and recurrence of points in terms of dynamical systems.

I want to understand why the following holds:

Let $(K,\varphi)$ a topological dynamical system and $(C(K),T_{\varphi})$ the Koopman system. Also for a closed set $A\subseteq K$ let $I_A:=\{f\in C(K): f_{\mid A}=0\}$. Then it holds that a point $x\in K$ is periodic if and only if $T^{n}_\varphi I_{\{x\}}= I_{\{x\}}$.

I know that $x\in K$ is called periodic if there is $n\in\mathbb{N}$ such that $\varphi^n(x)=x$. How can I use this information in the Koopman system and $T_\varphi$?

Answer 1

Let $x\in K$ be fixed and $(K;\varphi),T_\varphi$ be defined as in the book.

Part one:

If $n\in\Bbb N$ is such that $\varphi^n(x)=x$, then for any function $f$ defined on $K$, $T^n_\varphi(f)(x)=f(\varphi^n(x))=f(x)$ so $f(x)=0$ implies $T^n_\varphi(f)(x)=0$. Continuity or any topological considerations are irrelevant.

Part two:

If $n\in\Bbb N$ is such that $T^n_\varphi(I_{\{x\}})=I_{\{x\}}$, then for all continuous functions $f:K\to\Bbb R$, $f(x)=0\implies f(\varphi^n(x))=0$. Suppose for contradiction that $\varphi^n(x)=y\neq x$. As $K$ is Hausdorff, compact (the book defines its dynamical systems like this deliberately!) and therefore normal, there exists, by Urysohn’s lemma (this is maybe overkill but it works in the context of the book) a continuous function $f$ with $f(x)=0,\,f(y)=1$ ($\{y\},\{x\}$ are disjoint and closed) which means $f\notin T^n_\varphi(I_{\{x\}})$ but $f\in I_{\{x\}}$, a contradiction, so $\varphi^n(x)=x$ must hold.