The dynamo problem is considered for mappings with pulsed diffusion in the fast dynamo limit of vanishing magnetic diffusion. It is shown how the determinant of a dynamo operator may be expanded in terms of sums over the periodic orbits of the mapping. In mappings for which all the orbits are hyperbolic the limit of weak diffusion may be taken formally, yielding a prescription for calculating the fast dynamo growth rate from information about the periodic orbits. A mathematical justification for taking the limit of weak diffusion has not been obtained. Nevertheless it is verified that the prescription for calculating fast dynamo growth rates from periodic orbit sums gives correct growth rates for a number of models, including stretch-fold-shear and cat maps with shear.