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Topological conjugation classes of tightly transitive subgroups of Homeo₊(ℝ)

Enhui Shi, Lizhen Zhou (2016)

Colloquium Mathematicae

Let ℝ be the real line and let Homeo₊(ℝ) be the orientation preserving homeomorphism group of ℝ. Then a subgroup G of Homeo₊(ℝ) is called tightly transitive if there is some point x ∈ X such that the orbit Gx is dense in X and no subgroups H of G with |G:H| = ∞ have this property. In this paper, for each integer n > 1, we determine all the topological conjugation classes of tightly transitive subgroups G of Homeo₊(ℝ) which are isomorphic to ℤⁿ and have countably many nontransitive points.

Topological transitivity of solvable group actions on the line ℝ

Suhua Wang, Enhui Shi, Lizhen Zhou, Grant Cairns (2009)

Colloquium Mathematicae

Let ϕ:G → Homeo₊(ℝ) be an orientation preserving action of a discrete solvable group G on ℝ. In this paper, the topological transitivity of ϕ is investigated. In particular, the relations between the dynamical complexity of G and the algebraic structure of G are considered.

Transitive riemannian isometry groups with nilpotent radicals

C. Gordon (1981)

Annales de l'institut Fourier

Given that a connected Lie group G with nilpotent radical acts transitively by isometries on a connected Riemannian manifold M , the structure of the full connected isometry group A of M and the imbedding of G in A are described. In particular, if G equals its derived subgroup and its Levi factors are of noncompact type, then G is normal in A . In the special case of a simply transitive action of G on M , a transitive normal subgroup G ' of A is constructed with dim G ' = dim G and a sufficient condition is given...

Truncated Lie groups and almost Klein models

Georges Giraud, Michel Boyom (2004)

Open Mathematics

We consider a real analytic dynamical system G×M→M with nonempty fixed point subset M G. Using symmetries of G×M→M, we give some conditions which imply the existence of transitive Lie transformation group with G as isotropy subgroup.

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