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Invariant Varieties of Periodic Points, pp. 85-139 $100.00
Authors:  (Satoru Saito, Hakusan, Midori-ku, Yokohama, Japan, Noriko Saitoh, Dept. of Applied Mathematics, Yokohama National Univ., Hodogaya-ku, Yokohama, Japan)
When a map is given, there has not been known a way, just by investigation of the first few steps of its iterations, to predict if it is integrable or not. The purpose of this chapter is to provide such a method in the case of higher dimensional maps. We consider a d dimensional rational map with p invariants and study the behaviour of
periodic points in Cd. There are d − p independent periodicity conditions for each period, which determine periodic points isolated each other in general. It might happen, however, that they do not fix the periodic points but determine relations among the invariants. In this case the periodic points form a subvariety in Cd specified by
the invariants. We call this variety ‘an invariant varity of periodic points’, or IVPP for short. The IVPP is a discrete map analog of an invariant torus. We prove the theorem: “If there is an IVPP of some period, there is no set of isolated periodic points of any period in the map.” It is well known that the existence of isolated repulsive periodic points is a source of chaotic orbits which are sensitive to their initial points. On the other hand, by studying many known integrable maps, we find IVPP different for each period. Every point on one of the varieties is an initial point of a periodic map of the
same period. Therefore our theorem strongly suggests that the existence of an IVPP of some period is a sufficient condition for the map being integrable. A study of the nature of IVPP provides us useful information to expose the notion of integrability of higher dimensional maps. For example, by the restriction of the variables to an IVPP, the map becomes a recurrence map, such that all solutions are periodic for all initial points with the same period. As a map is perturbed, isolated periodic points are induced to the variety and the nature of the variety is changed significantly. We will discuss how the transition to a nonintegrable map takes place. 

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Invariant Varieties of Periodic Points, pp. 85-139