# 京都大学大学院 情報学研究科 数理工学専攻 応用数学講座 数理解析分野中村・辻本研究室

## 関西可積分系セミナー (2003年7月25日)

2003年7月25日（金）15時-17時

##### 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.