EVENT DETAILS AND ABSTRACT

Title: Algebraic dynamics from topological and holomorphic dynamics
Speaker: Rohini Ramadas
Speaker Info: Harvard University
Brief Description:
Special Note: Midwest Dynamical Systems Conference
Abstract:

Let f:S^2 —> S^2 be an orientation-preserving branched covering from the 2-sphere to itself whose postcritical set P := { f^n(x) | x is a critical point of f and n > 0 } is finite. Thurston studied the dynamics of f using an induced holomorphic self-map T(f) of the Teichmuller space of complex structures on (S^2, P). Koch found that this holomorphic dynamical system on Teichmuller space descends to an algebraic dynamical system. In particular, when P contains a point x at which f is fully ramified, under certain combinatorial conditions on f, the inverse of T(f) descends to a meromorphic self-map M(f) of projective space CP^n. When, in addition, x is a fixed point of f, i.e. f is a `topological polynomial’, the induced self-map M(f) is holomorphic.

The dynamics of M(f) may be studied via numerical invariants called dynamical degrees: the k-th dynamical degree of an algebraic dynamical system measures the asymptotic growth rate, under iteration, of the degrees of k-dimensional subvarieties. I will introduce the dynamical systems T(f) and M(f), and dynamical degrees. I will then discuss why the dynamical degrees of M(f) are algebraic integers, and how their properties are constrained by the dynamics of f on the finite set P. In particular, when M(f) exists, then the more f resembles a topological polynomial, the more M(f) behaves like a holomorphic map.