Discrete Dynamical Systems From Real Valued Mutation

Abstract

We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the two-dimensional situation.

Keywords:

• Cluster algebras
• integrable systems
• tropical dynamics

Acknowledgments

We would also like to thank Michael Shapiro for helpful comments on the content and exposition of the paper, and the anonymous reviewer for helpful comments and suggestions.

Additional information

Funding

N. Ovenhouse was supported by the NSF grant DMS-1745638.