### Geometry and Dynamics Seminar

# Sergei Duzhin

### (Steklov Institute St. Petersburg and MPIM Bonn)

## "Iterations of rational maps of the plane"

** Date: ** |
Tue, February 24, 2009 |

** Time: ** |
14:00 |

** Place: ** |
Research I Seminar Room |

**Abstract:** In this talk no theorems will be stated or proved.
We will only discuss some examples and give the results of
numerical experiments.

Let f be a real rational function of one variable.
Consider the map (x,y) -> (y,f(y)/x) of the plane into itself.
For some functions f, e.g. f(y)=1+y, this map is periodic.
For some others, e.g. f(y)=y+y^{-1}, it is in a certain sense close
to periodic.

The last mapping has surprising properties. Its iterations f, f^2,...,f^85
have only one fixed point in the positive quaterplane, while its 86-th
iteration has 172 extra fixed points, arranged along a curve which looks
fairly smooth, but is indeed fractal. To establish this fact numerically,
one needs to perform computations with at least 100 digits.

We will also discuss some rational maps of the non-commutative plane into
itself.