2015/11/23 by Inês Cruz, Cruz, Inês, Helena Mena-Matos +3
Chemistry · Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Molecular spectroscopy and chirality #math.DS
paper · pdf · doi:10.48550/arxiv.1511.07291
Proposition 4 removed due to partially incorrect statement. Remaining results are not affected
arxiv created 2015/12/22 · arxiv updated 2015/12/23
The dynamics of a 1-parameter family of cluster maps φr 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,ℤ)\ltimesℝ2. We conclude that there are three different types of dynamical behaviour for φr 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.