Random permutations of N elements have peculiar statistics of the lengths of the cycles (discovered by V.L. Gontcharov, 1942). The number of the cycles of a random permutation of N points grows with N (in the mean) as 1+ 1/2 +1/3 + … + 1/N ~ 0,58 + ln N. The Fibonacci cat map is the action of the matrix [ ] permuting the n^2 points of the finite torus Z/nZ x Z/nZ. Such algebraically defined permutations have quite different statistics (from that of the random ones). Example: Some permutations of 100 points have periods exceeding 230 million. The length of the period of the cat map, permuting 150x150 pixels, is only 300. The study of similar examples (begun in the 1990s by I. Persival and F. Dyson) has led to unexpected mathematical theorems. Example: The number of those permutations of 2N points, all whose cycles are of even length, is a square of an integer, namely, of (2N-1)!! =1.3.5…..(2N-1).
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