Linear Algebra and Topology
نویسندگان
چکیده
Two real entried (respectively: orthogonal) n x n matrices^! andB are said to be linearly similar if there is an invertible real (resp. orthogonal) n x n matrix C with CAC~ = B. Of course, A, B and C may be regarded as linear functions from R (resp. isometries from S~) to itself. The matrices A and B are said to be (resp. homogeneously differentiably) differentiably similar if there is a diffeomorphism ƒ: R —» R9 with 2 /(O) = 0 (resp. ƒ: S"" * S~) for which fAf = B. Differentiating this equation at 0, one sees immediately that differentiable similarity on R is the same as linear similarity with C = (Df)(0y A celebrated theorem of de Rham [7], [8] proves the corresponding equivalence for homogeneous differential similarity of orthogonal transformations of S ~ *. The matrices A and B are said to be (resp. homogeneously) topologically similar if there is a homeomorphism ƒ : R —> R with /(O) = 0 (resp. ƒ : 5«i —y S ~) and fAf~ = 5. For S , Poincaré showed, by defining rotation numbers, the equivalence of topological and linear similarity there. The study of the analogous question on S~ was continued by de Rham. In studying when topological similarity would imply Unear similarity on R, Kuiper and Robbin [5] showed that the general problem could be reduced to the following conjecture.
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تاریخ انتشار 2007