## Finnur Lárusson

Valentina Giangreco, júní 7, 2019

Málstofa í stærðfræði

### 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.