Search results for "Negative curvature"

showing 3 items of 3 documents

Prescribing the behaviour of geodesics in negative curvature

2010

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon desc…

Mathematics - Differential GeometryhoroballsPure mathematicsGeodesicDisjoint setsLagrange spectrum52A5501 natural sciences53C22Mathematics - Metric Geometry0103 physical sciences0101 mathematicshoroball[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]MathematicsFinite volume methodHall rayAMS : 53 C 22 11 J 06 52 A 55 53 D 25Mathematics - Number Theory010102 general mathematicsnegative curvatureRiemannian manifold[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]Closed geodesic53D25[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]Totally geodesic010307 mathematical physicsGeometry and TopologyNegative curvatureMathematics::Differential GeometryConvex functiongeodesicgeodesics11J06
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Counting common perpendicular arcs in negative curvature

2013

Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ from $D^-$ to $D^+$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^-$ and $D^+$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic form…

Mathematics - Differential GeometryGeneral Mathematics[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]37D40 37A25 53C22 30F4001 natural sciencesDomain (mathematical analysis)Bowen-Margulis measurecommon perpendicularequidistributiondecay of correlation0502 economics and businessortholength spectrummixingAsymptotic formulaSectional curvatureTangent vectorMathematics - Dynamical Systems0101 mathematicsExponential decayskinning measurelaskeminenMathematicsconvexityApplied Mathematicsta111010102 general mathematics05 social sciencesMathematical analysisRegular polygonnegative curvatureRiemannian manifoldGibbs measureManifoldKleinian groups[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]countingMathematics::Differential Geometrygeodesic arc050203 business & management
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Rate of Mixing for Equilibrium States in Negative Curvature and Trees

2021

In this survey based on the recent book by the three authors, we recall the Patterson-Sullivan construction of equilibrium states for the geodesic flow on negatively curved orbifolds or tree quotients, and discuss their mixing properties, emphasizing the rate of mixing for (not necessarily compact) tree quotients via coding by countable (not necessarily finite) topological shifts. We give a new construction of numerous nonuniform tree lattices such that the (discrete time) geodesic flow on the tree quotient is exponentially mixing with respect to the maximal entropy measure: we construct examples whose tree quotients have an arbitrary space of ends or an arbitrary (at most exponential) grow…

Pure mathematicssymbols.namesakeExponential growthDiscrete time and continuous timeThermodynamic equilibriumsymbolsCountable setNegative curvatureGibbs measureQuotientMathematicsExponential function
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