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From: aboulang@bbn.com (Albert Boulanger)
Newsgroups: comp.theory.dynamic-sys
Subject: Re: integer dynamical systems?
Message-ID: <ABOULANG.91Jun29142000@taurus.bbn.com>
Date: 29 Jun 91 18:20:00 GMT
References: <POLLACK.91Jun25121703@dendrite.cis.ohio-state.edu>
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In-reply-to: pollack@dendrite.cis.ohio-state.edu's message of 25 Jun 91 17:17:03 GMT

In article <POLLACK.91Jun25121703@dendrite.cis.ohio-state.edu> pollack@dendrite.cis.ohio-state.edu (Jordan B Pollack) writes:

   Are there any deterministic functions over the integers which are
   aperiodic?  Such a function would have to traverse an infinite subset
   of the integers in an "unpredictable" way, without halting or cycling.
   "Get the next prime number" would seem to be a candidate but for its
   monotonicity...


Interesting question, Jordan. Such a dynamical system whould have to
use large integers like normal chaotic ones use the fact that there is
"plenty of room at the bottom" to do its folding. One of the criteria
for chaos is ergodicity or that the orbits have to be dense. You seem
to have a conflicting desire to have it spread out as much as
possible. Anyway a good book on integer dynamical systems is:

Discrete Iterations
Francois Robert
Springer Verlag, 1986

Also there is an intersting n^.5 scaling of the size transient path
before these systems fall into their terminal cycle or fixed point.

For more information on this, see:

"The Simulation of Random Processes on Digital Computers: Unavoidable
Order" T. Erber & T Rynne, J of Comp Phys, 49, 394-419 (1983)

For possible quantum mechanical implications, see:

"Randomness in Quantum Mechanics -- Nature's Ultimate Cryptogram?"
T. Erber & S. Putterman, Nature, Vol 318 (7 Nov), 41-43, (1985)


Shifting my bits using a Mixmaster (TM),

Albert Boulanger
aboulanger@bbn.com