Seminar

Department of Mathematics

Monday, March 31st, 2008 at 3PM

BA6183, 40 St. George Street

DYNAMICAL SYSTEMS SEMINAR -- Smooth conjugacy of Anosov systems on higher dimensional tori.

A. Gogolev

Penn State University

Structural stability asserts that if two Anosov diffeomorphisms are close enough then they are conjugate: $hf=gh$. It's known that the conjugacy $h$ is Holder continuous. There are simple obstructions for $h$ to be smooth. Let $p$ be a periodic point of $f$, $f^n(p)=p$, $g^n(h(p))=h(p)$. If $h$ were differentiable, then the differentials $D(f^n)(p)$ and $D(g^n)(h(p))$ would be conjugate by the differential of $h$, and we say that $f$ and $g$ have same periodic data.

Question: Suppose that p. d. coincide, is h differentiable? If it is then how smooth is it?

This question was fully answered in dimension two by de la Llave, Marco and Moriyon (87-90).

We study smooth conjugacy problem for Anosov systems $C^1$-close to hyperbolic automorphism of higher dimensional tori.