#### Vol. 6, No. 1, 2013

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Addresses Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print)
Iterations of quadratic polynomials over finite fields

### William Worden

Vol. 6 (2013), No. 1, 99–112
##### Abstract

Given a map $f:ℤ\to ℤ$ and an initial argument $\alpha$, we can iterate the map to get a finite forward orbit modulo a prime $p$. In particular, for a quadratic map $f\left(z\right)={z}^{2}+c$, 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 $p\le x$ as $x\to \infty$, converges to the limiting distribution of ${X}_{n}∕\sqrt{n}$, as $n\to \infty$.

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