# Graduate School in Nonlinear Science

### Sponsored by the Danish Research Academy

**
MIDIT OFD CATS
Modelling, Nonlinear Dynamics Optics and Fluid Dynamics Chaos and Turbulence Studies
and Irreversible Thermodynamics Risø National Laboratory Niels Bohr Institute and
Technical University of Denmark Building 128 Department of Chemistry
Building 321 P.O. Box 49 University of Copenhagen
DK-2800 Lyngby DK-4000 Roskilde DK-2100 Copenhagen Ø
Denmark Denmark Denmark
**

**REAL QUADRATIC FAMILY: A COMPLETE PICTURE**

The main problem of dynamics is to describe a typical behaviour of orbits
for a typical system. It turned out to be a very hard problem even for
innocently looking families like the real quadratic family. However, the
following recent result gives a solution to the problem: Almost any real
quadratic map has either an attracting cycle, or an absolutely continuous
invariant measure. In the former ("regular") case, almost all orbits
converge to the cycle. In the latter ("stochastic") case, almost all
orbits are asymptotically equidistributed with respect to the measure. The
proof of this result incorporates all the theory mentioned above, and more.