Anosov automorphisms of nilpotent Lie algebras
Tracy L. Payne
Journal of Modern Dynamics 2009, 3(1): 121-158 doi: 10.3934/jmd.2009.3.121
Each matrix $A$ in $GL_n(Z)$ naturally defines an automorphism $f$ of the free $r$-step nilpotent Lie algebra $\frf_{n,r}$. We study the relationship between the matrix $A$ and the eigenvalues and rational invariant subspaces for $f$. We give applications to the study of Anosov automorphisms.
keywords: Anosov automorphism nilpotent Lie algebra. nilmanifold hyperbolic automorphism Anosov diffeomorphism Anosov Lie algebra
The Ricci flow for nilmanifolds
Tracy L. Payne
Journal of Modern Dynamics 2010, 4(1): 65-90 doi: 10.3934/jmd.2010.4.65
We consider the Ricci flow for simply connected nilmanifolds. This translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow.
keywords: nilmanifold Ricci flow soliton metric.

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