Arnold's cat picture

Arnold's cat image (also Anosov's cat image ) is in the theory of dynamic systems the simplest example of an Anosov diffeomorphism and thus an explicitly computable chaotic system . It is named after Vladimir Igorewitsch Arnold , who demonstrated the properties of transformation using the representation of a cat.

The picture shows how the illustration with the matrix deforms the unit square and how the pieces are rearranged modulo 1. The dashed lines indicate the directions of maximum expansion and compression, they correspond to the eigenvectors of the matrix. ${\displaystyle \left({\begin{matrix}2&1\\1&1\end{matrix}}\right)}$

definition

Arnold's cat image is defined by the self-image of the torus ${\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}}$

${\displaystyle F(x,y)=(2x+y,x+y)\mod 1}$

or in matrix notation

${\displaystyle F\left({\begin{bmatrix}x\\y\end{bmatrix}}\right)={\begin{bmatrix}2&1\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}\mod 1={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&0\\1&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}\mod 1.}$

properties

Discretization for n  = 150. After 300 iterations, the identity mapping is obtained again .

• The figure is an Anosov diffeomorphism: the matrix has two eigenvalues and , the eigenvectors provide a decomposition ${\displaystyle \left({\begin{matrix}2&1\\1&1\end{matrix}}\right)}$ ${\displaystyle \lambda _{1}>1}$ ${\displaystyle \lambda _{2}<1}$

${\displaystyle T_{x}T^{2}=E_{x}^{u}\oplus E_{x}^{s}}$

at each point , with and after the canonical identification ${\displaystyle x\in T^{2}}$ ${\displaystyle E_{x}^{u}}$ ${\displaystyle E_{x}^{s}}$

${\displaystyle T_{x}T^{2}\cong \mathbb {R} ^{2}}$

correspond to the eigenvectors and . The projections of the straight lines parallel to the eigenvectors onto the torus are the stable and unstable manifolds of the mapping. ${\displaystyle \lambda _{1}}$ ${\displaystyle \lambda _{2}}$

• The zero point is the only fixed point . The number of periodic points with period is ${\displaystyle n}$

${\displaystyle \lambda _{1}^{n}+\lambda _{2}^{n}-2}$.

The periodic points are close together. A point is pre-periodic if and only if it has rational coordinates.

• The mapping is topologically transitive .
• The mapping is area-conserving , ergodic and mixing .
• The reverse mapping is given by

${\displaystyle F^{-1}(x,y)=(2x-y,-x+y)\mod 1}$.

• The discretization

${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{2}\to (\mathbb {Z} /n\mathbb {Z} )^{2}}$

is periodic with period . ${\displaystyle \leq 3n}$

literature

• Vladimir I. Arnold , André Avez: Ergodic problems of classical mechanics. Translated from the French by A. Avez. WA Benjamin, Inc., New York - Amsterdam 1968.
• Freeman Dyson , Harold Falk: Period of a discrete cat mapping. Amer. Math. Monthly 99: 603-614 (1992).

Web links

• Eric W. Weisstein : Arnold's cat picture . In: MathWorld (English).