Arnold's Cat Map

The best known example of an Anosov diffeomorphism. It is given by the transformation

(1)

where and are computed mod 1. The Arnold cat mapping is non-Hamiltonian, nonanalytic, and mixing. However, it is area-preserving since the determinant is 1. The Lyapunov characteristic exponents are given by

(2)

so

(3)

The eigenvectors are found by plugging into the matrix equation

(4)

For , the solution is

(5)

where is the golden ratio, so the unstable (normalized) eigenvector is

(6)

Similarly, for , the solution is

(7)

so the stable (normalized) eigenvector is

(8)