Twist Map

A class of Area-Preserving Maps of the form

$\displaystyle \theta_{i+1}$	$\textstyle =$	$\displaystyle \theta_i+2\pi\alpha(r_i)$
$\displaystyle r_{i+1}$	$\textstyle =$	$\displaystyle r_i,$

which maps Circles into Circles but with a twist resulting from the $\alpha=\alpha(r_i)$ term.