Any bijection u : Z → → Z partitions the integers into a (possibly infinite) list of (possibly infinite) cycles: for each integer i, the cycle containing i is the sequence ( … … , u − − 2 ( i ) , u − − 1 ( i ) , i , u ( i ) , u 2 ( i ) , … … ) where exponentiation represents functional composition. For an affine permutation u, the following conditions are equivalent: all cycles of u are finite, u has finite order, and the geometric action of u on the space V has at least one fixed point.