Arnold Cat Map Digital Recurrence

A scrambled pixel picture can return perfectly because the digital cat map is a finite permutation.

Arnold Cat Map Digital Recurrence

Introduction

The Arnold Cat Map is a chaotic map from the torus into itself. Its discrete, digital analog is a finite permutation of points on an N×N grid. Given a transformation matrix A = [[2, 1], [1, 1]] applied modulo N, the mapping becomes (x, y) → (2x + y, x + y) mod N. Because det(A) = 1, it forms a bijection (a one-to-one correspondence) across the N*N grid. Thus, iterating the map will eventually return any point to its initial state, demonstrating exact digital recurrence.

Worked Example

Consider the grid modulus N = 5 and the initial point (1, 0), which is the "Axis". We want to trace its orbit under the Arnold Cat Map transformation modulo 5.

Step 0: Initial point is (x, y) = (1, 0).
Step 1: Apply A to (1, 0).
x' = (2(1) + 0) mod 5 = 2
y' = (1(1) + 0) mod 5 = 1
Point is (2, 1).
Step 2: Apply A to (2, 1).
x' = (2(2) + 1) mod 5 = 5 mod 5 = 0
y' = (1(2) + 1) mod 5 = 3 mod 5 = 3
Point is (0, 3).
Continuing this process, the orbit visits (3,3), (4,1), (4,0), (3,4), (0,2), (2,2), (1,4), and finally at step 10 returns to (1,0). The period is exactly 10.

Experiment

Trace the recurrence of points under the digital map in the interactive module athttps://riano.app/module/arnold-cat-map-digital-recurrence.

Set the Grid modulus to 5 and the Initial point to "Axis".
Set Iterations to 2. Observe the resulting point is exactly (0, 3) as computed.
Check the listed Point Period, which shows 10, meaning 10 steps are required for this point to return home.
Change the Initial point to "Origin". Notice that the period is 1, as (0,0) maps to (0,0).
Increase the Grid modulus to 8 or 10 and observe how the maximum period changes based on the grid size and initial point.

Limitations

A finite digital permutation is eventually periodic from the start. Although it visually scrambles an image (or a grid), it is not a true aperiodic chaotic system. Therefore, the apparent scrambling does not prove finite-grid aperiodic chaos and should not be relied upon for image encryption security. The continuous properties (expansion eigenvalue and Lyapunov exponent) are theoretical limit behaviors on the real continuous torus.

References

Ergodic Theory - Charles Walkden
Discussion of the Arnold Cat Map, its eigenvalues, and its chaotic properties on the continuous torus.

Arnold Cat Map Digital Recurrence