🗺️ Anosov Map — Arnold Cat Map & Hyperbolic Chaos
The Arnold cat map is the canonical example of an Anosov diffeomorphism — a hyperbolic area-preserving map on the torus: (x′, y′) = (x + y, x + 2y) mod 1. Every point is a hyperbolic fixed/periodic point: nearby trajectories diverge exponentially along the unstable manifold and converge equally fast along the stable manifold. Place any image on the torus: each step mixes it towards apparent randomness. But because the map is invertible and the torus has finitely many integer lattice points, any image returns exactly to its original state after a finite period (usually 1–200 steps for N≤256).
Image source
Map parameters
Animate
Stats
[x′; y′] = [[1,1];[1,2]]·[x;y] mod 1
General Anosov:
[x′; y′] = [[1,p];[q,pq+1]]·[x;y] mod 1
Eigenvalues:
λ = ½(tr ± √(tr²−4))
ln(λ₊) = Lyapunov exp.
Hyperbolic Dynamics & the Horseshoe
An Anosov diffeomorphism is hyperbolic everywhere — the tangent space at every point splits into stable (contracting) and unstable (expanding) subspaces. This makes the dynamics structurally stable: small perturbations do not change the qualitative character. The Arnold cat map is closely related to Smale's horseshoe map, which is the archetype of hyperbolic chaos. The fact that a chaotic map can be exactly reversed (quantum recurrence) makes it a valuable model in quantum information: it has been used to demonstrate quantum chaos and decoherence on quantum computers (Georgeot & Shepelyansky, 2001).