Arnold's cat map

In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name.^[1] It is a simple and pedagogical example for hyperbolic toral automorphisms.

Thinking of the torus ${\displaystyle \mathbb {T} ^{2}}$ as the quotient space ${\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}}$, Arnold's cat map is the transformation ${\displaystyle \Gamma :\mathbb {T} ^{2}\to \mathbb {T} ^{2}}$ given by the formula

${\displaystyle \Gamma (x,y)=(2x+y,x+y){\bmod {1}}.}$

Equivalently, in matrix notation, this is

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

That is, with a unit equal to the width of the square image, the image is sheared one unit up, then two units to the right, and all that lies outside that unit square is shifted back by the unit until it is within the square.