The eventual image


Venue   Nord Pas de Calais/Belgium Congress of Mathematics, Université de Mons, 30 October 2013.

Abstract   An endomorphism T of an object can be viewed as a discrete-time dynamical system: perform one iteration of T with every tick of the clock. This dynamical viewpoint suggests questions about the long-term destiny of the points of our object. (For example, does every point eventually settle into a periodic cycle?)

A fundamental concept here is the "eventual image", a companion to the algebraic concepts of image and kernel. Under suitable hypotheses, it can be defined as the intersection of the images of all the iterates Tn of T. I will explain its behaviour in three contexts: one set-theoretic, one algebraic, and one geometric. I will also present a unifying categorical framework, showing that the eventual image has not one but two universal properties, dual to one another. In this, it resembles other important constructions such as direct sum in an abelian category.

Slides   In this pdf file.

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