Complex dynamics perspective for birational maps of the plane arising from cluster algebra mutations

Abstract

Using the methods of holomorphic dynamics we investigate planar birational mappings that arise from the theory of cluster algebras and integrable systems. Computing dynamical degrees of these mappings, many of which are greater than one, allows us to show that many of the mappings do not have a conserved quantity (nor an invariant fibration). In most of the examples, invariant fibrations can also be ruled out by finding superattracting periodic points. This answers a question posted by Machacek and Ovenhouse 2024 and by Chen and Li 2024. Moreover, having found a good algebraically stable model for the mappings and having computed the dynamical degree, we can then apply results from the ergodic theory of birational maps to produce invariant measures with positive entropy and positive Lyapunov exponents.