Discrete-time Dynamical System: Integrable Or Not Integrable?

Abstract:

To any discrete-time dynamical system with a rational evolution, it is possible to associate a number, the "algebraic entropy", which is a global index of complexity of the system.

Its vanishing is the hallmark of integrability, and it is proved to be an unsurpassed detector of integrability. It is, however, also an interesting object in itself. Its analysis links in particular to algebraic geometry and number theory.

We will give the definition of the entropy, and show by simple examples how it can be calculated. We will then explain the link between this concept and the singularity structure of the system, relating it to the discrete Painleve analysis (this is where algebraic geometry comes in). We finally describe, in a simple case, the set of values assumed by the entropy (this is where number theory enters, as it is conjectured to always be the logarithm of an algebraic integer), and introduce the notion of entropy gap.