The nature of manifolds of periodic points for higher dimensional integrable maps

TL;DR

This paper investigates the structure of periodic points in higher-dimensional integrable maps, revealing conditions under which they form invariant varieties or isolated points, and proving their mutual exclusivity.

Contribution

It provides a detailed analysis of periodic point structures in rational maps with invariants, introducing criteria for their formation and proving their exclusive existence.

Findings

Invariant varieties have dimension p when periodicity conditions are fully correlated.

Periodic points are isolated when conditions are uncorrelated.

Invariant varieties and isolated points cannot coexist in the same map.

Abstract

By studying periodic points for rational maps on $\bm{C}^d$ with $p$ invariants, we show that they form an invariant variety of dimension $p$ if the periodicity conditions are `fully correlated', and a set of isolated points if the conditions are `uncorrelated'. We present many examples of the invariant varieties in the case of integrable maps. Moreover we prove that an invariant variety and a set of isolated points do not exist in one map simultaneously.