Vladlen Timorin: Dynamics of quadratic Thurston rational functions.

A rational function of one complex variable is a very simple algebraic object: a ratio of two polynomials. We will consider degree 2 (quadratic) rational functions f as dynamical systems. This means that, instead of just studying f(z), we will look at the sequence z, f(z), f(f(z)), …, called the orbit of z. How the behavior of the orbit depends on z and on f are the principal questions. We say that f is a Thurston rational function if the orbits of all critical points of f are (pre)periodic. Roughly, the role Thurston rational functions play in the world of all rational functions is similar to the role rational numbers play in the world of all real numbers. We will study quadratic Thurston rational functions through combinatorial objects associated with them.