Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!sun-barr!olivea!spool.mu.edu!sdd.hp.com!zaphod.mps.ohio-state.edu!cis.ohio-state.edu!dendrite.cis.ohio-state.edu!pollack From: pollack@dendrite.cis.ohio-state.edu (Jordan B Pollack) Newsgroups: comp.theory.dynamic-sys Subject: integer dynamical systems? Message-ID: Date: 25 Jun 91 17:17:03 GMT Sender: news@cis.ohio-state.edu (NETnews ) Reply-To: pollack@cis.ohio-state.edu Distribution: comp Organization: Ohio State Computer Science Lines: 28 Originator: pollack@dendrite.cis.ohio-state.edu Interestingly, Herman Rubin says of the (integer) hailstone sequences: >>The chaotic nature is that a slight difference at one point >>makes a big difference a considerable time in the future. Most of the widely discussed "chaotic" dynamical systems, like 4x(1-x), iterate over real numbers. But real numbers are not computable, and fractal images (presented by their proponents) are always floating point approximations to a platonic ideal. I find the "chaos" idea, of an iterative system having a "deterministic aperiodicity", quite stimulating precisely because it does not seem computable. I played with the hailstone numbers for a couple of hours once, and found them somewhat "unpredictable". It seemed to me that the sequences are the result of a non linear dynamical system over the integers (x -> x/2 if even, x -> 3x+1 if odd). But the hailstone system, though sensitive to initial parameter, always seems to halt by finding the (1-4-2-1) attractor, though I'm not sure this has been proven. Other non-linear operations, like modular multiplication by a constant, (the basis for pseudo-randomness), also lead to simple periodic cycles. 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...