A note on discrete dynamical systems in theories of class $S$
In this note we consider the set of line operators in theories of class $S$. We show that this set carries the action of a natural discrete dynamical system associated with the BPS spectrum. We discuss several applications of this perspective; the relation with global properties of the theory; the set of constraints imposed on the spectrum generator, in particular for the case of SU(2) $\mathcal{N}=2^*$; and the relation between line defects and certain spherical Double Affine Hecke Algebras.