Dynamics and periodicity in a family of cluster maps

Abstract

The dynamics of a 1-parameter family of cluster maps ${\phi }_r$ associated to mutation-periodic quivers in dimension 4, is studied in detail. The use of presymplectic reduction leads to a globally periodic symplectic map, and this enables us to reduce the problem to the study of maps belonging to a group of symplectic birational maps of the plane which is isomorphic to $SL(2,\mathbb{Z})⋉{\mathbb{R}}^2$. We conclude that there are three different types of dynamical behaviour for ${\phi }_r$ characterized by the integer parameter values $r=1$, $r=2$ and $r>2$. For each type, the periodic points, the structure and the asymptotic behaviour of the orbits are completely described. A finer description of the dynamics is provided by using first integrals.

Proposition 4 removed due to partially incorrect statement. Remaining results are not affected

References in corpus (2)

• Discrete integrable systems and Poisson algebras from cluster maps
• Dynamics of the birational maps arising from $F_0$ and $d{P}_3$ quivers