Arnold's cat map is inherently a 2D map on the unit torus:
(x′,y′) = (x+py, qx+(pq+1)y) mod 1

Eigenvalues λ± = ½(tr ± √(tr²−4)), tr = 2+pq.
Because pixel coords are finite integers, the map is a permutation — it always returns to the original image after a finite period.