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Concept# Arnold's cat map

Summary

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 as the quotient space , Arnold's cat map is the transformation given by the formula
Equivalently, in matrix notation, this is
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.
Γ is invertible because the matrix has determinant 1 and therefore its inverse has integer entries,
Γ is area preserving,
Γ has a unique hyperbolic fixed point (the vertices of the square). The linear transformation which defines the map is hyperbolic: its eigenvalues are irrational numbers, one greater and the other smaller than 1 (in absolute value), so they are associated respectively to an expanding and a contracting eigenspace which are also the stable and unstable manifolds. The eigenspaces are orthogonal because the matrix is symmetric. Since the eigenvectors have rationally independent components both the eigenspaces densely cover the torus. Arnold's cat map is a particularly well-known example of a hyperbolic toral automorphism, which is an automorphism of a torus given by a square unimodular matrix having no eigenvalues of absolute value 1.
The set of the points with a periodic orbit is dense on the torus. Actually a point is periodic if and only if its coordinates are rational.
Γ is topologically transitive (i.e. there is a point whose orbit is dense).
The number of points with period is exactly (where and are the eigenvalues of the matrix). For example, the first few terms of this series are 1, 5, 16, 45, 121, 320, 841, 2205 .... (The same equation holds for any unimodular hyperbolic toral automorphism if the eigenvalues are replaced.)
Γ is ergodic and mixing,
Γ is an Anosov diffeomorphism and in particular it is structurally stable.

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