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RESEARCH PRODUCT

Hasse diagrams and orbit class spaces

Gioia M. VagoH. HattabChristian BonattiE. Salhi

subject

Pure mathematicsMathematical analysisOrbit classClosure (topology)Hasse diagramTopological spaceGroup of homeomorphismsQuotient space (linear algebra)Hasse principleRealizationHomogeneous spaceCovering relationFinitely generated groupGeometry and TopologyHasse diagramMathematics

description

Abstract Let X be a topological space and G be a group of homeomorphisms of X. Let G ˜ be an equivalence relation on X defined by x G ˜ y if the closure of the G-orbit of x is equal to the closure of the G-orbit of y. The quotient space X / G ˜ is called the orbit class space and is endowed with the natural order inherited from the inclusion order of the closure of the classes, so that, if such a space is finite, one can associate with it a Hasse diagram. We show that the converse is also true: any finite Hasse diagram can be realized as the Hasse diagram of an orbit class space built from a dynamical system ( X , G ) where X is a compact space and G is a finitely generated group of homeomorphisms of X.

yearjournalcountryeditionlanguage
2011-04-01Topology and its Applications
10.1016/j.topol.2010.12.010http://dx.doi.org/10.1016/j.topol.2010.12.010