Dynamics on Algebraic Groups

Abstract

Using the Artin-Mazur dynamical zeta function, we study the periodic behavior of discrete dynamical systems arising from algebraic groups over algebraically closed fields of positive characteristic. Of particular interest are maps arising as finite quotients of affine morphisms on algebraic groups; so-called dynamically affine maps. We present a set of hypotheses that imply that the corresponding dynamical zeta function is either a root of a rational function, or has a natural boundary. This generalizes recent work by Bridy for dynamically affine maps on the projective line. Under slightly weaker assumptions, we show that the tame dynamical zeta function, formed by ignoring orbits whose order is divisible by the characteristic, is always a root of a rational function. We work towards a conjectural description of the orbit structure for endomorphisms of arbitrary connected algebraic groups. This extends recent work by Byszewski and Cornelissen for the case of an abelian variety.