arXiv subjects
Publications and source records attributed to Noriko Saitoh.
We show that, when a non-integrable rational map changes to an integrable one continuously, a large part of the Julia set of the map approach indeterminate points (IDP) of the map along algebraic curves. We will see that the IDPs are singular loci of the curves.
The behaviour of periodic points of discrete Euler top is studied. We derive invariant varieties of periodic points explicitly. When the top is axially symmetric they are specified by some particular values of the angular velocity along the axis of symmetry, different for each period.
A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic points of higher dimensional integrable maps.
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 which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}
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.
We study periodicity conditions of a rational map on $\bm{C}^d$ with $p$ invariants and show that a set of isolated periodic points and an algebraic variety of finife dimension do not exist in one map simultaneously if $p\ge d/2$. We also discuss in detail how the transition takes place between them.
We propose a method to characterize discrete time evolution equations, which generalize discrete time soliton equations, including the $q$-difference Painlevé IV equations discussed recently by Kajiwara, Noumi and Yamada.
We study a small piece of two dimensional Toda lattice as a complex dynamical system. In particular the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.
A difference analogue of the logistic equation, which preserves integrability, is derived from Hirota's bilinear difference equation. The integrability of the map is shown to result from the large symmetry associated with the Bäcklund transformation of the KP hierarchy. We introduce a scheme which interpolates between this map and the standard logistic map and enables us to study integrable and nonintegrable systems on an equal basis. In particular we study the behavior of Julia set at the point where the nonintegrable map passes to the integrable map.