Chaotic torus automorphism and Poincaré recurrence
Image
Statistics
Iteration:0
Period (N=60):60
Lyapunov exp:ln(φ²)≈0.962
Eigenvalues:φ², 1/φ²
About
[x,y] → [2x+y, x+y] mod 1. Matrix {{2,1},{1,1}} has det=1 (area-preserving) and eigenvalues φ²≈2.618 and φ⁻²≈0.382 where φ=(1+√5)/2. The map is hyperbolic and ergodic. For an N×N grid, it is periodic with period dividing 3N for prime N. Around iteration 60 for N=60 the image nearly returns (not exact for this N).