The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps
Misael Avendaño-Camacho 1,, , Isaac Hasse-Armengol 2, , Eduardo Velasco-Barreras 1,3, and Yury Vorobiev 1,
Departamento de Matemáticas, Universidad de Sonora, Blvd. Luis Encinas y Rosales, s/n, Col. Centro, C.P. 83000, Hermosillo, Son., México
Departamento de Física y Matemáticas, Universidad Autónoma de Ciudad Juárez, Av. del Charro no. 450 nte., Col. Partido Romero, C.P. 32310, Ciudad Juárez, Chihuahua, México
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Niterói 24210-201, Río de Janeiro, Brasil
* Corresponding author: Misael Avendaño-Camacho
Received September 2019 Revised February 2020 Published June 2020
Fund Project: The authors are supported by CONACYT grant CB2015 no. 258302. E. Velasco - Barreras was supported by FAPERJ grants E-26/202.411/2019 and E-26/202.412/2019
On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.
Keywords: Poisson foliation, cotangential group action, invariant Ehresmann connections, averaging method, pre-momentum map, Dirac structure.
Mathematics Subject Classification: Primary: 53D17, 53C12; Secondary: 53D20, 58D19, 70G45.
Citation: Misael Avendaño-Camacho, Isaac Hasse-Armengol, Eduardo Velasco-Barreras, Yury Vorobiev. The method of averaging for Poisson connections on foliations and its applications. Journal of Geometric Mechanics, 2020, 12 (3) : 343-361. doi: 10.3934/jgm.2020015
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Misael Avendaño-Camacho Isaac Hasse-Armengol Eduardo Velasco-Barreras Yury Vorobiev