A combinatorial model for reversible rational maps over finite fields

TL;DR

This paper analyzes the cycle structure of reversible rational maps over finite fields, proposing a probabilistic model involving involutions that explains observed symmetries and cycle length distributions.

Contribution

It introduces a novel combinatorial model using involutions to describe cycle statistics of reversible maps over finite fields, confirming conjectured distributions.

Findings

Cycle lengths follow a universal distribution R(x)=1-e^{-x}(1+x).

Most cycles are asymptotically symmetrical.

Cycle repetition probabilities follow a Poisson law.

Abstract

We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e., map-independent) distribution function R(x)=1-e^{-x}(1+x) has been conjectured to exist, for the normalized cycle lengths of the reduced map in the large field limit (J. A. G. Roberts and F. Vivaldi, Nonlinearity 18 (2005) 2171-2192). We show that these statistics correspond to those of a composition of two random involutions, having an appropriate number of fixed points. This model also explains the experimental observation that, asymptotically, almost all cycles are symmetrical, and that the probability of occurrence of repeated periods is governed by a Poisson law.