Arnold's cat map
（重定向自Cat map）
![Picture showing how the linear map stretches the unit square and how its pieces are rearranged when the modulo operation is performed. The lines with the arrows show the direction of the contracting and expanding eigenspaces](/Images/godic/202501/03/Arnoldcatmap.svg5644.png")
![](/Images/godic/202501/03/Arnold_cat5644.png")
5644.gif")
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.
Thinking of the torus ![\mathbb{T}^2](/Images/godic/202501/03/28c7d18676fc1bff34c1ff8c5dfe19a75644.png")
as the quotient space ![\mathbb{R}^2/\mathbb{Z}^2](/Images/godic/202501/03/62f1174ceff7bf1b1bc52a3040d5c6465644.png")
Arnold's cat map is the transformation ![\Gamma : \mathbb{T}^2 \to \mathbb{T}^2](/Images/godic/202501/03/32ea25d7337c0a3920d9e11fb5ecf4245644.png")
given by the formula