Arnold cat map - Knowlpedia

The cat map is the automorphism of $\mathbb T^2$ given by

T_A(x)=Ax\bmod\mathbb Z^2,\qquad A=\begin{pmatrix}2&1\\1&1\end{pmatrix}.

Its determinant is one and its inverse has integer entries, so it is an invertible group automorphism preserving Haar probability. It is strongly mixing.

Fourier mechanism

For $e_k(x)=e^{2\pi ik\cdot x}$ ,

U_{T_A}e_k=e_{A^{\mathsf T}k},\qquad \langle U_{T_A}^ne_k,e_\ell\rangle=1_{\{(A^{\mathsf T})^nk=\ell\}}.

The eigenvalues of $A$ are $(3\pm\sqrt5)/2$ . Neither eigenline contains a nonzero integer vector. Every nonzero integer $k$ therefore has a nonzero expanding component and $\|(A^{\mathsf T})^nk\|\to\infty$ .

Each displayed correlation is eventually zero for fixed nonzero frequencies. Density of trigonometric polynomials proves mixing for all mean-zero $L^2$ observables. These Fourier characters form a useful basis, but they are not Koopman eigenfunctions: the operator permutes their frequencies.