Invariant varieties of periodic points for some higher dimensional integrable maps

TL;DR

This paper demonstrates that in certain higher-dimensional integrable maps, periodic points form invariant varieties of dimension at least equal to the number of invariants, contrasting with nonintegrable maps where such points are isolated.

Contribution

It proves a theorem linking invariant varieties of periodic points to the absence of isolated periodic points of different periods in integrable maps.

Findings

Periodic points form invariant varieties of dimension ≥ p

Invariant varieties exist for each period in integrable maps

Nonintegrable maps have isolated periodic points

Abstract

By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}