Arnold Cat Map Digital Recurrence

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.

Experiment

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

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