Hyperbolic Behavior

and Hyperbolic Spaces

Professor Keen

Conformal dynamical systems arise as models of various physical,
economic and social phenomena. A dynamical system may be described by
a family of functions, f(z,c), depending on two variables. For each
fixed value of c we are interested in the orbits of a given point z;
that is, one considers the set of points z, f(c,z), f(f(c,z)), and so
on. One particular family of functions Professor Keen has studied
is f(c,z)= c tan(z). So, for a particular function f(1,z) the orbit
of a point z is {z, tan(z), tan(tan(z)), tan(tan(tan(z)))... } but the
orbit of f(0, z) is just {z, 0}.

If a small change in the input data, z, changes the orbit only slightly, we say the system is stable, or predictable. If a small change in z, on the other hand, causes a drastic change in the orbit, we say the system is chaotic. For most stable systems the orbits tend to a periodic orbit. If, for a stable system, a small change in the parameter c changes the limiting periodic orbit slightly, the system is called hyperbolic. It would seem that hyperbolic behavior is the norm, but this is often difficult to prove. Professor Keen has proved that hyperbolic behavior occurs for a special class of functions and is in the process of extending her result to a larger class of functions.

Professor Keen is also researching a class of 3 dimensional spaces called hyperbolic 3-manifolds. They are so-called because they are endowed with a non-Euclidean geometry called a hyperbolic geometry. In such a geometry the sum of the angles in a triangle is less than a 180 degrees. The hyperbolic 3-manifold may be varied so that the geometry changes, but remains hyperbolic. In previous work, based on careful analysis of examples generated by sophisticated computer programs, Prof Keen has found a good way to describe one and two parameter families of hyperbolic 3-manifolds. She is extending this work to families depending on a arbitrary number of parameters.

A webpage with art based on Kleinian Groups and hyperbolic geometry.

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