Finnur Lárusson

Valentina Giangreco, júní 7, 2019

Málstofa í stærðfræði

Fyrirlesari: Finnur Lárusson, Adelaide University

Titill: Chaotic holomorphic automorphisms of Stein manifolds with the 
volume density property

Staðsetning: VR-II, V-158
Tími: þriðjudagur 9. júlí kl. 11.00

Ágrip:

I will report on joint work with Leandro Arosio.  Let $X$ be 
a Stein manifold of dimension $n\geq 2$ satisfying the volume density 
property with respect to an exact holomorphic volume form.  For example, 
$X$ could be $\C^n$, any connected linear algebraic group that is not 
reductive, the Koras-Russell cubic, or a product $Y\times\C$, where $Y$ 
is any Stein manifold with the volume density property. We prove that 
chaotic automorphisms are generic among volume-preserving holomorphic 
automorphisms of $X$.  In particular, $X$ has a chaotic holomorphic 
automorphism. Forn\ae ss and Sibony proved (but did not explicitly 
state) this for $X=\C^n$ in 1997.  We follow their approach closely. 
Peters, Vivas, and Wold showed that a generic volume-preserving 
automorphism of $\C^n$, $n\geq 2$, has a hyperbolic fixed point whose 
stable manifold is dense in $\C^n$.  This property can be interpreted as 
a kind of chaos.  We generalise their theorem to a Stein manifold as above.