Fixed Point Homing Shuffles

We study a family of maps from Sₙ → Sₙ we call fixed point homing shuffles. These maps generalize a few known problems such as Conway's Topswops, and a card shuffling process studied by Gweneth McKinley. We show that the iterates of these homing shuffles always converge, and characterize the set Uₙ of permutations that no homing shuffle sorts. We also study a homing shuffle that sorts anything not in Uₙ, and find how many iterations it takes to converge in the worst case. Updated formatting to fit with DMTCS requirements