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Dynamics and periodicity in a family of cluster maps
- Publication Year :
- 2015
-
Abstract
- The dynamics of a 1-parameter family of cluster maps $\varphi_r$ 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 $\varphi_r$ 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.<br />Comment: Proposition 4 removed due to partially incorrect statement. Remaining results are not affected
- Subjects :
- Mathematics - Dynamical Systems
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1511.07291
- Document Type :
- Working Paper