Invariant subsets of rank 1 manifolds

Buyalo, Sergei; Schroeder, Viktor

doi:10.1007/s002290100225

Cited by 3 publications

(2 citation statements)

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“…Theorem B2 is motivated by the unpublished work of Burns and Pollicott [6] and subsequent papers [8,35,7,32], where hyperbolic manifolds and more general manifolds of nonpositive curvature are considered. However, in all the aforementioned papers the set Ξ consisted of a single point.…”

Section: 2mentioning

confidence: 99%

Nondense orbits on homogeneous spaces and applications to geometry and number theory

An¹,

Guan²,

Kleinbock³

2020

Preprint

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Let G be a Lie group, Γ ⊂ G a discrete subgroup, X = G/Γ, and f an affine map from X to itself. We give conditions on a submanifold Z of X guaranteeing that the set of points x ∈ X with f -trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.

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Section: 2mentioning

confidence: 99%

Nondense orbits on homogeneous spaces and applications to geometry and number theory

An¹,

Guan²,

Kleinbock³

2020

Preprint

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“…The next generalization took place in [3]. Buyalo and Schroeder considered a compact manifold of rank one and proved the existence of a neighbourhood like W for every vector of rank one.…”

Section: Introductionmentioning

confidence: 99%

Flow invariant subsets for geodesic flows of manifolds with non-positive curvature

Reinold¹

2004

Ergod. Th. Dynam. Sys.

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Consider a closed, smooth manifold M of non-positive curvature. Write p:UM→M for the unit tangent bundle over M and let > denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow on UM. We define the structured dimension s-dim > which, essentially, is the dimension of the set p( > ) of base points of > . The main result of this paper holds for manifolds with s-dim > 0, there is an -dense, flow invariant, closed subset Ξ UM > such that p(Ξ )=M. Abstract. Consider a closed, smooth manifold M of non-positive curvature. Write p : UM → M for the unit tangent bundle over M and let R > denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow φ on UM. We will define the structured dimension s-dim R > which, essentially, is the dimension of the set p(R > ) of base points of R > .The main result of this paper holds for manifolds with s-dim R > < dim M/2: for every > 0, there is an -dense, flow invariant, closed subset ⊂ UM\R > such that p( ) = M.

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Non-dense orbits on homogeneous spaces and applications to geometry and number theory

An¹,

Guan²,

Kleinbock

2021

Ergod. Th. Dynam. Sys.

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Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

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Invariant subsets of rank 1 manifolds

Cited by 3 publications

References 0 publications

Nondense orbits on homogeneous spaces and applications to geometry and number theory

Nondense orbits on homogeneous spaces and applications to geometry and number theory

Flow invariant subsets for geodesic flows of manifolds with non-positive curvature

Non-dense orbits on homogeneous spaces and applications to geometry and number theory

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