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!
Shuffling cards and the horseshoe map
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reference-request
dynamical-systems
markov-chains
chaos-theory
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0Thanks, t.b.! I just requested Shub's book via Interlibrary Loan. – 2011-12-02
1 Answers
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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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0Nice observation re stripping off a bit! Thanks, Ross! – 2011-12-02