An Equidistribution Result in Non-Archimedean Dynamics

Series: 
Algebra Seminar
Monday, January 26, 2015 - 15:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
University of Georgia
Organizer: 
Let K be a complete, algebraically closed, non-Archimedean field, and let $\phi$ be a rational function defined over K with degree at least 2.  Recently, Robert Rumely introduced two objects that carry information about the arithmetic and the dynamics of $\phi$. The first is a function $\ord\Res_\phi$, which describes the behavior of the resultant of $\phi$ under coordinate changes on the projective line. The second is a discrete probability measure $\nu_\phi$ supported on the Berkovich half space that carries arithmetic information about $\phi$ and its action on the Berkovich line. In this talk, we will show that the functions $\ord\Res_\phi(x)$ converge locally uniformly to the Arakelov-Green's function attached to $\phi$, and that the family of measures $\nu_{\phi^n}$ attached to the iterates of $\phi$ converge to the equilibrium measure of $\phi$.‚Äč