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I wonder if there is a connection between the dynamics of repeated cut & shuffle operations on a deck of cards, and topological chaotic maps such as the horseshoe map? I ask this entirely naively. Pointers to where I could explore such a connection would be appreciated—Thanks!

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I'm not sure if that is what you're looking for, but one such connection is provided by symbolic dynamics. If I remember correctly, a nice introduction to this is in Shub's book Global stability of dynamical systems. –  t.b. Dec 2 '11 at 13:42
    
Thanks, t.b.! I just requested Shub's book via Interlibrary Loan. –  Joseph O'Rourke Dec 2 '11 at 13:54

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You can think of shuffling as doubling. So if the card position starts out $n \in [0,25]$, an out-shuffle sends it to $2n$. If it starts out $n \in [26,51]$, an out-shuffle sends it to $2n-1 \pmod {52}=2n-53$. This is similar to the chaotic system that just strips the first bit off a number.

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Nice observation re stripping off a bit! Thanks, Ross! –  Joseph O'Rourke Dec 2 '11 at 18:29

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