The nature of manifolds of periodic points for higher dimensional   integrable maps
  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. 
Satoru Saito, Noriko Saitoh