Global periodicity conditions for maps and recurrences via Normal Forms

TL;DR

This paper develops a method using Normal Form Theory and rational parametrization to identify parameter values that produce periodic behavior in families of rational maps and recurrences, simplifying the detection of such cases.

Contribution

It introduces a novel approach combining Normal Form Theory and rational parametrization to characterize global periodicity in parametric rational maps and recurrences.

Findings

Identified parameter sets for periodic maps in various dimensions.

Applied method to Lyness type recurrences to find global periodic cases.

Reduced complex periodicity detection to simple algebraic computations.

Abstract

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences.