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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 whichthey are isolated. We prove the theorem: {\it `If there is an invariant varietyof periodic points of some period, there is no set of isolated periodic pointsof other period in the map.'}Comment: 24 page

Invariant Varieties of Periodic Points for Some Higher Dimensional Integrable Maps