Vol. 6, No. 1, 2013

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Iterations of quadratic polynomials over finite fields

William Worden

Vol. 6 (2013), No. 1, 99–112

Given a map f : and an initial argument α, we can iterate the map to get a finite forward orbit modulo a prime p. In particular, for a quadratic map f(z) = z2 + c, where c is constant, work by Pollard suggests that the forward orbit should have length on the order of p. We give a heuristic argument that suggests that the statistical properties of this orbit might be very similar to the birthday problem random variable Xn, for an n = p day year, and offer considerable experimental evidence that the limiting distribution of the orbit lengths, divided by p, for p x as x , converges to the limiting distribution of Xnn, as n .

arithmetic dynamics, birthday problem, forward orbit modulo $p$, random maps
Mathematical Subject Classification 2010
Primary: 37P05
Secondary: 11B37
Received: 2 March 2012
Revised: 25 April 2012
Accepted: 10 May 2012
Published: 23 June 2013

Communicated by Michael Zieve
William Worden
Temple University
Wachman Hall Rm. 517
1805 N. Broad St.
Philadelphia, PA 19122
United States