Fate of the Julia set of higher dimensional maps in the integrable limit

By studying higher dimensional rational maps, we have shown, in our previous papers, that periodic points of integrable maps with sufficient number of invariants form invariant varieties of periodic points (IVPPs) different for each period. In this paper, we study the transition of a nonintegrable map to an integrable one. In particular, we investigate analytically where the Julia set goes and how it disappears when the map becomes integrable. We show that the behavior of the Julia set is different, depending on whether the map has an unstable variety of fixed point (UVFP), which becomes nonfixed in the integrable limit. If the map does not have an UVFP, all periodic points approach IVPP or fixed points of the integrable map. Otherwise, a large part of periodic points of all periods approach the UVFP and the UVFP itself becomes a variety of indeterminate points unless it disappears from the map. Moreover, we show that a map recovered by the singularity confinement generates the sequence of all IVPPs.