$f_\alpha:S^1\rightarrow S^1, \alpha\geq 2$ be given by$ f_\alpha (\theta)=2\theta, \forall \theta \in [0,2\pi]$ prove f is chaotic?

Question

$f_\alpha:S^1\rightarrow S^1, \alpha\geq 2$ be given by$ f_\alpha (\theta)=2\theta, \forall \theta \in [0,2\pi]$ prove f is chaotic?

Now I know I need to prove that 1) F is topologically transitive 2) the orbits of f are dense

It seems trivial that the orbits of f are dense (correct me if i'm wrong) but how do I prove that F is topologically transitive?

Thanks

Answer 1

Each time you apply $f$ to an open interval in $S^1$ it doubles its length. If $(a,b)\subset S^1$ is an open interval in the circle, then there is an $n\in\mathbb{N}$ such that $f^n((a,b))=S^1$.