I am studying the dynamics of the map $f(z)=z^2$ on $\overline{\mathbb{C}}$, where $\overline{\mathbb{C}}$ is Riemann sphere, from the book of Alan Beardon "Iteration of rational maps" . I want to study the dynamics of this map on the unit circle $S^1$. Since under this map the circle $S^1$ is completely invariant ,thus, I have \begin{align} f(z): S^1 \rightarrow S^1\\ e^{2 \pi i \theta} \mapsto e^{2 \pi i 2\theta} \end{align}
It is written in the book that we can ignore the integral part of $2\theta$ as exponential function is periodic with period $2\pi i$. It follows , then, that we can understand the action of $f$ on $S^1$ if we understand the action of the map \begin{align} g: [0,1) \rightarrow [0,1)\\ \theta \mapsto 2\theta \;\; (mod \;1) \end{align} on the interval $[0,1)$.
I didn't understand how this follow from the periodicity of exponential function. Why does this map sufficient to study? Is it because map $f(z): S^1 \rightarrow S^1$ given by $e^{2 \pi i \theta} \mapsto e^{2 \pi i 2\theta}$ topologically conjugate to the map $g$? If yes then what is the conjugacy map here?