# Dynamics of the birational maps arising from F0 and dP3 quivers

@article{Cruz2015DynamicsOT, 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}, year={2015}, volume={431}, pages={903-918} }

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…

## 3 Citations

Dynamics and periodicity in a family of cluster maps

- Mathematics
- 2015

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

- Mathematics
- 2016

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

- Physics
- 2020

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…

## References

SHOWING 1-10 OF 17 REFERENCES

Symplectic Maps from Cluster Algebras

- Mathematics, Physics
- 2011

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

- Mathematics
- 2010

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

- Mathematics
- 2006

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

- Mathematics, Physics
- 2014

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

- Mathematics
- 2014

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

- Mathematics
- 1996

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

- Mathematics, Physics
- 2011

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

- Mathematics
- 2010

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

- Physics
- 1988

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

- Mathematics, Physics
- 2002

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.