Iterations of quadratic polynomials over finite fields

Worden, William

doi:10.2140/involve.2013.6.99

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Iterations of quadratic polynomials over finite fields

William Worden

Vol. 6 (2013), No. 1, 99–112

DOI: 10.2140/involve.2013.6.99

Abstract

Given a map $f : ℤ \to ℤ$ 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 $f (z) = z^{2} + c$ , where $c$ is constant, work by Pollard suggests that the forward orbit should have length on the order of $\sqrt{p}$ . We give a heuristic argument that suggests that the statistical properties of this orbit might be very similar to the birthday problem random variable $X_{n}$ , for an $n = p$ day year, and offer considerable experimental evidence that the limiting distribution of the orbit lengths, divided by $\sqrt{p}$ , for $p \leq x$ as $x \to \infty$ , converges to the limiting distribution of $X_{n} ∕ \sqrt{n}$ , as $n \to \infty$ .

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Keywords

arithmetic dynamics, birthday problem, forward orbit modulo $p$, random maps

Mathematical Subject Classification 2010

Primary: 37P05

Secondary: 11B37

Milestones

Received: 2 March 2012

Revised: 25 April 2012

Accepted: 10 May 2012

Published: 23 June 2013

Communicated by Michael Zieve

Authors

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

Vol. 6, No. 1, 2013