Apr

23

Friday, April 30, 2010, 2-3 p.m., in BA 6183, 40 St. George St. PhD Candidate: Mario Morfin Ramirez PhD Advisors: Michael Shub and Charles Pugh Thesis title: Grassmann dynamics Thesis Abstract: The present work is divided in two parts. The first is concerned with the dynamics on the Grassmann manifold of k-dimensional subvector spaces of an n dimensional real or complex vector space induced by a linear invertible transformation A of the vector space into itself. In 1968 Steve Batterson attacked a similar problem in his Northwestern University doctoral dissertation. He considered the action induced by diagonal transformations whose eigenvalues have distinct moduli. He gives a topological classification of these maps and proves that they are Morse-Smale. In the current work, I considere the non diagonal case and give a complete topological classification of the conjugacy classes of the linearly induced maps on the Grassmann manifold. Also, I define an extension of the Morse-Smale diagrams to describe these conjugacy classes, and I discuss the structural stability and bifurcation properties among and within the conjugacy classes. In the second part, I consider dynamics induced by a linear cocycle covering a diffeomorphism f:N --> N of a compact manifold, acting on the Grassmann bundle of k-dimensional linear subspaces of T N. Two problems are attacked: A Kupka-Smale theorem for the space of cocycles covering diffeomorphisms of a compact manifold. The proof of this theorem implies the same type of results for derived cocycles parametrized in the space of diffeomorphisms. The results of the second part can be generalized without effort to cocycles covering endomorphisms of N.

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