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arXiv · 1511.07291

Dynamics and periodicity in a family of cluster maps

Inês Cruz·Helena Mena-Matos·M. Esmeralda Sousa-Dias
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Abstract

The dynamics of a 1-parameter family of cluster maps $φ_r$ 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 $SL(2,\mathbb{Z})\ltimes\mathbb{R}^2$. We conclude that there are three different types of dynamical behaviour for $φ_r$ 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.

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Inês Cruz, Helena Mena-Matos, M. Esmeralda Sousa-Dias. 2015-12-22. Dynamics and periodicity in a family of cluster maps. https://arxiv.org/abs/1511.07291

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