# Meeting Details

Title: A generalization of a Differentiable Linearization of Philip Hartman Working Seminar: Dynamics and its Working Tools Sheldon Newhouse, Michigan State University A linear automorphism L of $R^n$ is "bi-circular" if its spectrum lies in two disjoint circles $C_1, C_2$ in the complex plane such that the radius of $C_1$ is less than 1 and the radius of $C_2$ is greater than 1. A fixed point $p$ of $C^1$ diffeomorphism $f$ is "bi-circular" if the derivative $Df(p)$ is bi-circular. A well-known theorem of Philip Hartman says that a bi-circular fixed point $p$ of a $C^{1,1}$ diffeomorphism $f$ (i.e. the derivative map is Lipschitz near $p$) is $C^1$ linearizable near $p$. We extend this to the case in which $f$ is $C^{1,a}$ with $0 < a < 1$ (i.e., $Df$ is Holder continuous with exponent $a$). Our proof also works in the infinite dimensional case where $R^n$ is replaced by a real Banach space which has $C^{1,a}$ bump functions (e.g. Hilbert spaces). The results can be used to give simpler proofs under weaker differentiability assumptions of results of L. P. Shilnikov (and his collaborators) giving horseshoe type dynamics near certain homoclinic curves.