April 20, 2021
Title: Endomorphisms of algebraic groups from the viewpoint of dynamical systems
Speaker: Gunther Cornelissen
Consider the following two problems:
(a) count the number of irreducible polynomials over a finite field, asymptotically in thedegree of the polynomial (i.e., a polynomial analogue of the prime number theorem) and
(b) compute the number of invertible nxn matrices over a finite field.
These apparently different looking problems can both be interpreted in tems of dynamics of an endomorphism of an algebraic group (where the endomorphism is “Frobenius” and the algebraic group is the additive group or GL(n), and we count orbits, or fixed points). The talk is about a vast generalisation of these results to arbitrary endomorphisms of arbitrary algebraic groups with finitely many fixed points. The results will be explained by example and by picture.
Typical technical keywords are: adelic distortion of linear recurrent sequences, non-hyperbolicity in dynamical systems, Steinberg’s formula for reductive groups, rationality of Artin-Mazur zeta functions, analogue of the Riemann Hypothesis for orbit counts (ongoing joint work with Jakub Byszewski and Marc Houben).