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関西可積分系セミナー (2003年7月25日)

日時
2003年7月25日(金)15時-17時
場所
京都大学本部キャンパス工学部総合校舎406会議室
William A. Schwalm (University of North Dakota)
Finding Lie groups that reduce certain systems of difference equations

This is partly an informal talk about groups and difference equations, and partly a discussion of some strategies we have used to find relevant groups. Difference systems xm+1=f(xm), where each component of f: Rn→Rn is a rational function, appear frequently. Maeda showed that if G={gα} is a Lie group such that each gα commutes with f, or such that for each x and α, f(gα(x))=gα(f(x)), then the order of the system is reduced by transforming into canonical variables of G. In a difference system arising from a physical problem, one has f and needs to fine G. While Lie provided ways of finding groups in the continuous case of differential equations, we do not know of such a systematic approach in the finite difference case. Maeda studied the connection between dynamical invariants and symmetries in discrete mechanics. Quispel matched infinitesimal generators to Poincare expansions near fixed points of f. Another tool we have used is the relation between the fixed or invariant sets of f and of G. (An invariant set M of f is such that x∈M→f(x)∈M.) Especially in the case of renormalization recursions that come from lattice dynamics problems, which do not preserve measure, it is useful to study preimages of the invariant sets of f. One can develop necessary criteria, based on singularities, in order that a given difference system admit a subgroup of the projective linear group, or of certain quadratic Cremona Lie groups. We consider some simple examples to illustrate a few of the general ideas. One is a model system that reduces to a general logistic map, one is related to renormalization of Schrodinger's equation on a fractal, and one is related to a discrete Kepler problem.