Dynamics of the birational maps arising from F0 and dP3 quivers

  title={Dynamics of the birational maps arising from F0 and dP3 quivers},
  author={In{\^e}s Cruz and Helena Mena-Matos and M. Esmeralda Sousa-Dias},
  journal={Journal of Mathematical Analysis and Applications},
Abstract The dynamics of the maps associated to F 0 and d P 3 quivers is studied in detail. We show that the corresponding reduced symplectic maps are conjugate to globally periodic maps by providing explicit conjugations. The dynamics in R + N of the original maps is obtained by lifting the dynamics of these globally periodic maps and the solution of the discrete dynamical systems generated by each map is given. A better understanding of the dynamics is achieved by considering first integrals… 
Dynamics and periodicity in a family of cluster maps
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
Multiple Reductions, Foliations and the Dynamics of Cluster Maps
Reduction of cluster maps via presymplectic and Poisson structures is described in terms of the canonical foliations defined by these structures. In the case where multiple reductions coexist, we
The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps
We consider a family of birational maps \begin{document}$ \varphi_k $\end{document} in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach


Symplectic Maps from Cluster Algebras
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver,
Symplectic birational transformations of the plane
We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{SL}(2,\mathbb{Z})$, the torus and a special map of order $5$, as it was
Global periodicity and complete integrability of discrete dynamical systems
Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F p (x)=x for all x in the phase space. On the other
Discrete Integrable Systems and Poisson Algebras From Cluster Maps
We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified
Reduction of cluster iteration maps
We study iteration maps of difference equations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and presymplectic geometry, we show that these
k-integrals and k-Lie symmetries in discrete dynamical systems
We generalize the concept of symplectic maps to that of k- symplectic maps: maps whose kth iterates are symplectic. Similarly, k-symmetries and k-integrals are symmetries (resp. integrals) of the kth
Cluster mutation-periodic quivers and associated Laurent sequences
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the
Discrete Integrable Systems
10.1007/978-1-4419-9126-3 Copyright owner: Springer Science+Buisness Media, LLC, 2010 Data set: Springer Source Springer Monographs in Mathematics The rich subject matter in this book brings in
Integrable mappings and soliton equations
Abstract We report an 18-parameter family of integrable reversible mappings of the plane. These mappings are shown to occur in soliton theory and in statistical mechanics. We conjecture that all
Introduction to Dynamical Systems
This chapter discusses the dynamics of measure-theoretic entropy through the lens of anosov diffeomorphisms, a type of topological dynamics that combines topological and Symbolic dynamics.