Arnold's Cat Map

The best known example of an Anosov Diffeomorphism. It is given by the Transformation

$$\left[{\matrix{x_{n+1} \cr y_{n+1}\cr}}\right] = \left[{\mat... ...\cr 1 & 2 \cr}}\right]\left[{\matrix{x_n \cr y_n \cr}}\right],$$ (1)

where $x_{n+1}$ and $y_{n+1}$ 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

$$\left\vert\matrix{1-\sigma & 1 \cr 1 & 2-\sigma \cr}\right\vert = \sigma^2-3\sigma+1 = 0,$$ (2)

so

$$\sigma_\pm = {\textstyle{1\over 2}}(3\pm\sqrt{5}\,).$$ (3)

The Eigenvectors are found by plugging $\sigma_\pm$ into the Matrix Equation

$$\left[{\matrix{1-\sigma_\pm & 1 \cr 1 & 2-\sigma_\pm \cr}}\r... ...trix{x\cr y \cr}}\right] = \left[{\matrix{0\cr 0 \cr}}\right].$$ (4)

For $\sigma_+$, the solution is

$$y={\textstyle{1\over 2}}(1+\sqrt{5}\,)x\equiv \phi x,$$ (5)

where $\phi$ is the Golden Ratio, so the unstable (normalized) Eigenvector is

$$\boldsymbol{\xi}_+ = {\textstyle{1\over 10}}\sqrt{50-10\sqrt{5}}\,\left[{\matrix{1 \cr {1\over 2}(1+\sqrt{5}\,)\cr}}\right].$$ (6)

Similarly, for $\sigma_-$, the solution is

$$y=-{\textstyle{1\over 2}}(\sqrt{5}-1)x\equiv \phi^{-1} x,$$ (7)

so the stable (normalized) Eigenvector is

$$\boldsymbol{\xi}_- = {\textstyle{1\over 10}}\sqrt{50+10\sqrt{5}}\,\left[{\matrix{1 \cr {1\over 2}(1-\sqrt{5}\,)\cr}}\right].$$ (8)

See also Anosov Map