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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.19 no.2 Temuco jun. 2017

http://dx.doi.org/10.4067/S0719-06462017000200049 

Articles

On topological symplectic dynamical systems

S. Tchuiaga1 

M. Koivogui2 

F. Balibuno3 

V. Mbazumutima3 

1 The University of Buea, Department of Mathematics, South West Region, Cameroon. tchuiagas@gmail.com

2 Ecole Supérieure Africaine des Technologies de l'Information et de Communication, Côte d' Ivoire. moussa.koivogui@esatic.ci

3 Institut de Mathématiques et des Sciences Physiques Bénin. balibuno.lugando@imsp-uac.org, mbazumutima.vianney@aims-cameroon.org

Abstract:

This paper contributes to the study of topological symplectic dynamical systems, and hence to the extension of smooth symplectic dynamical systems. Using the positivity result of symplectic displacement energy (4), we prove that any generator of a strong symplectic isotopy uniquely determine the latter. This yields a symplectic analogue of a result proved by Oh (12), and the converse of the main theorem found in (6). Also, tools for defining and for studying the topological symplectic dynamical systems are provided: We construct a right-invariant metric on the group of strong symplectic homeomorphisms whose restriction to the group of all Hamiltonian homeomorphism is equivalent to Oh's metric (12), define the topological analogues of the usual symplectic displacement energy for non-empty open sets, and we prove that the latter is positive. Several open conjectures are elaborated.

Keywords and Phrases: Isotopies; Diffeomorphisms; Homeomorphisms; Displacement energy; Hofer-like norms; Mass flow; Riemannian metric; Lefschetz type manifolds; Flux geometry.

Resumen:

Este artículo contribuye al estudio de los sistemas dinámicos simplécticos topológicos, y por tanto a la extensión de los sistemas dinámicos simplécticos suaves. Usando el resultado de la positividad de la energía de desplazamiento simpléctico (4), demostramos que cualquier generador de una isotopía simpléctica determina esta última. Esto entrega un análogo simpléctico de un resultado demostrado por Oh (12), y el inverso del teorema principal encontrado en (6). También entregamos herramientas para definir y estudiar los sistemas dinámicos simplécticos topológicos: construimos una métrica invariante por derecha en el grupo de homeomorfismos fuertemente simplécticos cuya restricción al grupo de homeomorfismos Hamiltonianos es equivalente a la métrica de Oh (12), definimos los análogos topológicos de la energía de desplazamiento simpléctico usual para conjuntos no-vacíos, y demostramos que esta última es positiva. Planteamos varios problemas abiertos.

2010 AMS Mathematics Subject Classification: 53D05, 53D35, 57R52, 53C21.

Acknowledgments:

The first author thanks the African Academy of Sciences, the African Institute for Mathematical Sciences of South Africa, and the Chair of Mathematics of the African Institute for Mathematical Sciences of Senegal, for the financial support of his researches.

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