Canonical Representations of the Simple Map

The simple map is the simplest map that has the topology of a divertor tokamak. The simple map has three canonical representations: (i) toroidal flux and poloidal angle ($\psi $,$\theta)$ as canonical coordinates, (ii) the physical variables (R,Z) or (X,Y) as canonical coordinates, and (iii) the action-angle (J,$\zeta)$ or magnetic variables ($\Psi $,$\Theta)$ as canonical coordinates. We give the derivation of the simple map in the (X,Y) representation. The simple map in this representation has been studied extensively (Ref. 1 and references therein). We calculate the magnetic coordinates for the simple map, construct the simple map in magnetic coordinates, and calculate generic topological effects of magnetic perturbations in divertor tokamaks using the map. We also construct the simple map in ($\psi $,$\theta)$ representation. Preliminary results of these studies will be presented. This work is supported by US DOE OFES DE-FG02-01ER54624 and DE-FG02-04ER54793. \newline [1] A. Punjabi, H. Ali, T. Evans, and A. Boozer, \textit{Phys Lett A} \textbf{364} 140--145 (2007).