## Finnur Lárusson

Math Colloquium

### Speaker: Finnur Lárusson, Adelaide University

### Title: Chaotic holomorphic automorphisms of Stein manifolds with the

volume density property

Location: VR-II, V-158

Time: Tuesday July 9 at 11.00 am

### Abstract:

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.