Structurally stable Grassmann transformations

Batterson, Steve

doi:10.1090/S0002-9947-1977-0448443-6

Structurally stable Grassmann transformations
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Trans. Amer. Math. Soc. 231 (1977), 385-404 Request permission

Abstract:

A Grassmann transformation is a diffeomorphism on a Grassmann manifold which is induced by an $n \times n$ nonsingular matrix. In this paper the structurally stable Grassmann transformations are characterized to be the maps which are induced by matrices whose eigenvalues have distinct moduli. There is exactly one topological conjugacy class of complex structurally stable Grassmann transformations. For the real case the topological classification is determined by the ordering (relative to modulus) of the signs of the eigenvalues of the inducing matrix.

References

Charles G. Cullen, Matrices and linear transformations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0197472
Charles Ehresmann, Sur la topologie de certains espaces homogènes, Ann. of Math. (2) 35 (1934), no. 2, 396–443 (French). MR 1503170, DOI 10.2307/1968440
John Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc. 158 (1971), 301–308. MR 283812, DOI 10.1090/S0002-9947-1971-0283812-3

Notes on manifolds of

mappings

M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc. 76 (1970), 1015–1019. MR 292101, DOI 10.1090/S0002-9904-1970-12537-X
Nicolaas H. Kuiper, Topological conjugacy of real projective transformations, Topology 15 (1976), no. 1, 13–22. MR 405508, DOI 10.1016/0040-9383(76)90046-X
N. H. Kuiper and J. W. Robbin, Topological classification of linear endomorphisms, Invent. Math. 19 (1973), 83–106. MR 320026, DOI 10.1007/BF01418922
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR 0649788
J. Palis and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 223–231. MR 0267603
J. W. Robbin, A structural stability theorem, Ann. of Math. (2) 94 (1971), 447–493. MR 287580, DOI 10.2307/1970766
J. W. Robbin, Topological conjugacy and structural stability for discrete dynamical systems, Bull. Amer. Math. Soc. 78 (1972), 923–952. MR 312529, DOI 10.1090/S0002-9904-1972-13058-1
Clark Robinson, Structural stability of $C^{1}$ diffeomorphisms, J. Differential Equations 22 (1976), no. 1, 28–73. MR 474411, DOI 10.1016/0022-0396(76)90004-8

Non-genericity of rational zeta functions and instability in

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0295244