  Chaotic dynamics is an important source for generating pseudorandom binary
sequences (PRNS). Much efforts have been devoted to obtaining period
distribution of the generalized discrete Arnold's Cat map in various domains
using all kinds of theoretical methods, including Hensel's lifting approach.
Diagonalizing the transform matrix of the map, this paper gives the explicit
formulation of any iteration of the generalized Cat map. Then, its real graph
(cycle) structure in any binary arithmetic domain is disclosed. The subtle
rules on how the cycles (itself and its distribution) change with the
arithmetic precision $e$ are elaborately investigated and proved. The regular
and beautiful patterns of Cat map demonstrated in a computer adopting
fixed-point arithmetics are rigorously proved and experimentally verified. The
results will facilitate research on dynamics of variants of the Cap map in any
domain and its effective application in cryptography. In addition, the used
methodology can be used to evaluate randomness of PRNS generated by iterating
any other maps.
