Sheldon Newhouse, Department of Astrophysical and Planetary Scienes (APS), University of Colorado Boulder
What we do and do not know about simple dynamical systems
It is well-known that simple dynamical systems can have complicated orbit structures. We describe some recent progress in understanding the structures of polynomial maps and other simple maps in one and two dimensional dynamics. In dimension one, there is now a fairly complete theory, while in dimension two there are related conjectures. At least in many two-dimensional cases it can be shown that there are invariant sets with dense orbits and maximal Hausdorff dimension. The methods have relations to old results on continued fractions due to Marshall Hall and apply to oscillatory motions in the three body problem.
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