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

TL;DR

This paper investigates the structure of periodic points in higher-dimensional integrable maps, revealing conditions under which isolated points and algebraic varieties coexist or transition, advancing understanding of their geometric properties.

Contribution

It establishes that in rational maps with at least half the dimension in invariants, isolated periodic points and algebraic varieties cannot coexist, and explores the transition between these cases.

Findings

Isolated periodic points and algebraic varieties do not coexist when p ≥ d/2.

Detailed analysis of the transition between isolated points and varieties.

Provides conditions for the existence and transition of periodic structures.

Abstract

We study periodicity conditions of a rational map on $\bm{C}^d$ with $p$ invariants and show that a set of isolated periodic points and an algebraic variety of finife dimension do not exist in one map simultaneously if $p\ge d/2$. We also discuss in detail how the transition takes place between them.