 

Annie S. Chen, T. Alden Gassert, and Katherine E. Stange
Index divisibility in dynamical sequences and cyclic orbits modulo p view print


Published: 
August 18, 2017

Keywords: 
arithmetic dynamics, dynamical portrait, index divisibility, cycle, orbit, functional digraph, dynamical sequence, polynomial map, iteration, quadratic map, divisibility sequence, integer sequence, postcritical orbit 
Subject: 
Primary: 37P05, 37P25, 11Y55, Secondary: 11B37, 11B39, 11B50, 11G99 


Abstract
Let ϕ(x) = x^{d} + c be an integral polynomial of degree at least 2, and consider the sequence (ϕ^{n}(0))_{n=0}^{∞}, which is the orbit of 0 under iteration by ϕ. Let D_{d,c} denote the set of positive integers n for which n  ϕ^{n}(0). We give a characterization of D_{d,c} in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes p have the property that the orbit of 0 is a single pcycle modulo p. We show that the set of such primes is finite when d is even, and conjecture that it is infinite when d is odd.


Acknowledgements
The third author's work was supported by the National Security Agency grant H982301610040 and National Science Foundation grant DMS1643552.


Author information
Annie S. Chen:
Stanford University, 450 Serra Mall, Stanford, CA 94305
asc8@stanford.edu
T. Alden Gassert:
Hobart and William Smith Colleges, 300 Pulteney Drive, Geneva, NY 14456
gassert@hws.edu
Katherine E. Stange:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 803090395
kstange@math.colorado.edu

