# Discrete Integrable System and Invariant Variety of Periodic Points

pp.1 - 128 , 2015-03-25

In this thesis, we discuss the nature of periodic points of discrete integrable systems. We consider, in particular, integrable rational maps and/or algebraic difference equations(ADE), whose behavior we can decide precisely for all initial conditions. It was shown, in the last ten years, that periodic points of such a system form an algebraic variety different for each period if the system has a sufficient number of invariants. Since every variety is determined only by information of the invariants it is called an invariant variety of periodic points(IVPP). It was also suggested that the existence of an IVPP might be sufficient to characterize integrability of rational maps. This is because we can prove that the coexistence of an IVPP and a discrete set of periodic points of any period is forbidden in one map. Thus an IVPP guarantees non existence of the Julia set, a fractal set of unstable periodic points which characterizes non-integrability of the system. Having studied the properties of IVPPs in many discrete maps we encountered various interesting phenomena common in such systems. For example, the algebraic varieties associated with IVPPs of different periods intersect each other. The intersections form a variety which is singular because every point of the variety is occupied by points of different periods simultaneously. The main purpose of this thesis is to explore where and how the intersections of IVPPs can take place. First by studying the periodicity conditions for the maps in detail we will arrive at a proposition that the intersections are possible only on the singularities of the maps. In the case of rational maps the zero set of denominators form a variety of singular points(SP), while the points satisfying 0/0 form a variety of indeterminate points(IDP). In the ADE case the IDPs can be determined from the implicit function theorem. Many integrable maps are investigated by means of computer algebra to confirm our proposition. Based on this observation we have found the following phenomena in this work.・Let us consider a d-dimensional map with p invariants. After elimination of p variables by using all the invariants, we obtain ADE for (d-p) variables whose coefficients are parameterized by the invariants. If we fix the parameters such that the invariants specify the IVPP of period n, the ADE becomes a recurrence equation of period n. Namely, all solutions of this ADE are n periodic for any initial point.・On the other hand if we parameterize the SP of a rational map by the invariants, any point on the SP must be mapped, after n steps, to the IVPP of period n. In other words, the singular points of the map are the source of IVPPs. This fact enables us to derive IVPPs of all periods iteratively. Moreover we will show that this phenomenon can be associated with a "projective resolution" of "triangulated category”.・Finally we investigate how the transition takes place between an integrable map and a non-integrable map. To this end we introduce an arbitrary parameter a to an integrable map, such that the map becomes integrable at a=0. When a is finite the repelling periodic points of the map form the Julia set. As a varies the value continuously every point of the Julia set moves along an algebraic curve. We see that some of them approach IVPPs in the a=0 limit but a large part of them approach the singular points of the integrable map, so that the points become singular loci of the algebraic curves which are highly degenerate for each period.

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