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Arnold's Cat Map

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Arnold's cat map is an Anosov diffeomorphism of the 2-torus T^2=R^2/Z^2. Using unit-square representatives, it is given by the transformation

[x_(n+1); y_(n+1)]=[1 1; 1 2][x_n; y_n],
(1)

where x_(n+1) and y_(n+1) are computed modulo 1. Write A for this matrix and F for the resulting map. It is area-preserving since detA=1. As a map of the torus, it is a real analytic diffeomorphism and a mixing transformation with respect to Lebesgue measure. The jumps in the unit-square representation come from the cuts used to represent the torus.

The induced action on the first homology group is A, so F is not isotopic to the identity map. In particular, it is not a Hamiltonian diffeomorphism of the torus, even though it preserves area.

The eigenvalues of A satisfy

|1-sigma 1; 1 2-sigma|=sigma^2-3sigma+1=0,
(2)

so

sigma_+/-=1/2(3+/-sqrt(5)).
(3)

The corresponding Lyapunov characteristic exponents are

lambda_+/-=lnsigma_+/-=+/-ln((3+sqrt(5))/2).
(4)

The eigenvectors are found by plugging sigma_+/- into the matrix equation

[1-sigma_+/- 1; 1 2-sigma_+/-][x; y]=[0; 0].
(5)

For sigma_+, the solution is

y=1/2(1+sqrt(5))x=phix,
(6)

where phi is the golden ratio, so the unstable (normalized) eigenvector is

xi_+=1/(10)sqrt(50-10sqrt(5))[1; 1/2(1+sqrt(5))].
(7)

Similarly, for sigma_-, the solution is

y=-1/2(sqrt(5)-1)x=-phi^(-1)x,
(8)

so the stable (normalized) eigenvector is

xi_-=1/(10)sqrt(50+10sqrt(5))[1; 1/2(1-sqrt(5))].
(9)
ArnoldCatMapConstruction

The figure above shows the unit square, its linear image under A, and the pieces wrapped back modulo 1. The colors identify corresponding pieces, with their boundaries drawn to show the cuts.

For a semialgebraic set S subset Q=[0,1)^2, its image can be calculated by stretching, cutting, and translating,

F(S)= union _(k in K)[(AS intersection (k+Q))-k],
(10)

where K={0,1}×{0,1,2} and AS is the linear image of S under A. Only these six integer translations are needed because AQ subset [0,2)×[0,3). Each piece is a semialgebraic set, so cylindrical algebraic decomposition can be used for exact iteration of such regions (Trott 2006b). Keeping the indicated endpoint conventions assigns points on the cuts exactly once.

For a positive integer N>=2, reducing integer coordinates modulo N instead gives a permutation of an N×N grid. The matrix powers are

A^k=[F_(2k-1) F_(2k); F_(2k) F_(2k+1)]
(11)

for k>=1, where F_j is a Fibonacci number (Dyson and Falk 1992, p. 604; Trott 2006a). The least period of this permutation is the smallest positive integer k for which A^k=I (modN), where I is the identity matrix.

ArnoldCatMapIterations

The figure above shows a cat image on a 202×202 grid after 0, 1, 2, 5, 10, and 75 iterations. The grid map with N=202 has least period 75, so the last panel reproduces the original image. Individual periodic points can have smaller least periods, as the origin is a fixed point.

See also

Anosov Diffeomorphism, Anosov Map

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References

Akritas, P.; Antoniou, I. E.; and Pronko, G. P. "On the Torus Automorphisms: Analytic Solution, Computability and Quantization." Chaos Solitons Fractals 12, 2805-2814, 2001. https://doi.org/10.1016/S0960-0779(01)00094-7.Arnold, V. I. and Avez, A. Ergodic Problems of Classical Mechanics. New York: Benjamin, 1968.Bokulich, A. "Horizontal Models: From Bakers to Cats." Philos. Sci. 70, 609-627, 2003. https://doi.org/10.1086/376927.Chen, G.; Mao, Y.; and Chui, C. K. "A Symmetric Image Encryption Scheme Based on 3D Chaotic Cat Maps." Chaos Solitons Fractals 21, 749-761, 2004. https://doi.org/10.1016/j.chaos.2003.12.022.Dyson, F. J. and Falk, H. "Period of a Discrete Cat Mapping." Amer. Math. Monthly 99, 603-614, 1992. https://doi.org/10.2307/2324989.Gaspari, G. "The Arnold Cat Map on Prime Lattices." Physica D 73, 352-372, 1994. https://doi.org/10.1016/0167-2789(94)90105-8.Georgeot, B. and Shepelyansky, D. L. "Stable Quantum Computation of Unstable Classical Chaos." Phys. Rev. Lett. 86, 5393-5396, 2001. https://doi.org/10.1103/PhysRevLett.86.5393.Georgeot, B. and Shepelyansky, D. L. "Quantum Computer Inverting Time Arrow for Macroscopic Systems." Eur. Phys. J. D 19, 263-266, 2002. https://doi.org/10.1140/epjd/e20020077.Trott, M. "Euler, Bernoulli, and Fibonacci Numbers." §2.4 in The Mathematica GuideBook for Numerics. New York: Springer-Verlag, 2006a. https://doi.org/10.1007/0-387-28814-7. https://www.mathematicaguidebooks.org/.Trott, M. "Polynomials in Inequalities." §1.2.3 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006b. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Arnold's Cat Map

Cite this as:

Weisstein, Eric W. "Arnold's Cat Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ArnoldsCatMap.html

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