Here's a passage of my script I do not understand.
Define $\Omega:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ and consider $a\in [0,1)$. Then $$ T_a:=\Omega\to\Omega, z\mapsto z\cdot \exp(2\pi ia) $$ defines a dynamical system.
Now my questions are:
(1) What exactly does $T_a$ do? If I have f.e. $a=0$ then $T_a(z)=z$.
(2) Why does $T_a$ define a dynamical system (definition below)? What is here $(X,\mathcal{A},\mathbb{P})$?
Our definition of dynamical system:
A quadrupel $(X,\mathcal{A},\mathbb{P},T)$ is called a dynamic system if $(X,\mathcal{A},\mathbb{P})$ is a probability space and $T\colon X\to X$ is a measurable transformation that holds the measure, i.e. $\mathbb{P}\circ T^{-1}=\mathbb{P}$.