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.