scholarly journals Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

2021 ◽  
Vol 17 (0) ◽  
pp. 401
Author(s):  
Dubi Kelmer ◽  
Hee Oh
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
General Theorem ◽  
Hyperbolic Manifolds ◽  
Quantitative Estimates ◽  
Geometrically Finite ◽  
Logarithm Law ◽  
Tubular Neighborhoods ◽  
Logarithm Laws ◽  
Metric Balls

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{M} $\end{document}</tex-math></inline-formula> be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.</p>

scholarly journals A note on Myrberg points and ergodicity

2005 ◽  
Vol 96 (1) ◽  
pp. 107 ◽  
Author(s):  
Kurt Falk
Keyword(s):  
Geodesic Flow ◽  
Kleinian Group ◽  
Hyperbolic Manifold ◽  
Strong Relationship ◽  
Hyperbolic Manifolds ◽  
Limit Set

The main purpose of this note is to further clarify the strong relationship between ergodicity and Myrberg type dynamics in hyperbolic manifolds. This is achieved mainly via a new suggestive proof of the fact that the Myrberg limit set of a Kleinian group is of full Patterson measure if the geodesic flow on the associated hyperbolic manifold is ergodic with respect to the Patterson-Sullivan measure.

scholarly journals Lagrangian systems on hyperbolic manifolds

1999 ◽  
Vol 19 (5) ◽  
pp. 1157-1173 ◽  
Author(s):  
PHILIP BOYLAND ◽  
CHRISTOPHE GOLÉ
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Invariant Sets ◽  
Lagrangian System ◽  
Mather Theory ◽  
Closed Surfaces ◽  
Bounded Distance ◽  
Time Periodic

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler–Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry–Mather theory of twist maps and the Hedlund–Morse–Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

scholarly journals On interpreting Patterson–Sullivan measures of geometrically finite groups as Hausdorff and packing measures

2015 ◽  
Vol 36 (8) ◽  
pp. 2675-2686
Author(s):  
DAVID SIMMONS
Keyword(s):  
New York ◽  
Finite Groups ◽  
Acta Math ◽  
Hyperbolic Manifolds ◽  
Packing Measure ◽  
Geometrically Finite ◽  
Logarithm Law ◽  
Conformal Measures ◽  
Finite Kleinian Group ◽  
Geometrically Finite Groups

We provide a new proof of a theorem whose proof was sketched by Sullivan [Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math.149(3–4) (1982), 215–237], namely that if the Poincaré exponent of a geometrically finite Kleinian group $G$ is strictly between its minimal and maximal cusp ranks, then the Patterson–Sullivan measure of $G$ is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of Stratmann [Multiple fractal aspects of conformal measures; a survey. Workshop on Fractals and Dynamics (Mathematica Gottingensis, 5). Eds. M. Denker, S.-M. Heinemann and B. Stratmann. Springer, Berlin, 1997, pp. 65–71; Fractal geometry on hyperbolic manifolds. Non-Euclidean Geometries (Mathematical Applications (N.Y.), 581). Springer, New York, 2006, pp. 227–247].

Ergodic behaviour of Sullivan's geometric measure on a geometrically finite hyperbolic manifold

1982 ◽  
Vol 2 (3-4) ◽  
pp. 491-512 ◽  
Author(s):  
Daniel J. Rudolph
Keyword(s):  
Ergodic Theorem ◽  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
Geometric Measure ◽  
Mean Ergodic Theorem ◽  
Geometrically Finite ◽  
Ergodic Behaviour

AbstractSullivan's geometric measure on a geometrically finite hyperbolic manifold is shown to satisfy a mean ergodic theorem on horospheres and through this that the geodesic flow is Bernoulli.

The displacement function and the return map

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Displacement Function ◽  
Semisimple Element ◽  
Fixed Point Set ◽  
Critical Set ◽  
Quantitative Estimates ◽  
Point Set ◽  
Toponogov’S Theorem

This chapter studies the displacement function dᵧ on X that is associated with a semisimple element γ‎ ∈ G. If φ‎″, t ∈ R denotes the geodesic flow on the total space X of the tangent bundle of X, the critical set X(γ‎) ⊂ X of dᵧ can be easily related to the fixed point set Fᵧ ⊂ X of the symplectic transformation γ‎⁻¹φ‎₁ of X. The chapter studies the nondegeneracy of γ‎⁻¹φ‎₁ − 1 along Fᵧ. More fundamentally, this chapter gives important quantitative estimates on how much φ‎ ½ differs from φ‎ ˗½γ‎ away from Fᵧ. These quantitative estimates are based on Toponogov's theorem.

scholarly journals Diastatic entropy and rigidity of complex hyperbolic manifolds

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Roberto Mossa
Keyword(s):  
Lower Bound ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Continuous Map ◽  
The Hyperbolic Metric ◽  
Geometric Rigidity ◽  
Volume Entropy ◽  
Complex Hyperbolic Manifold ◽  
Real Analytic

AbstractLet f : Y → X be a continuous map between a compact real analytic Kähler manifold (Y, g) and a compact complex hyperbolic manifold (X, g0). In this paper we give a lower bound of the diastatic entropy of (Y, g) in terms of the diastatic entropy of (X, g0) and the degree of f . When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary,when X = Y,we get that the minimal diastatic entropy is achieved if and only if g is isometric to the hyperbolic metric g0.

scholarly journals Maximally stretched laminations on geometrically finite hyperbolic manifolds

2017 ◽  
Vol 21 (2) ◽  
pp. 693-840 ◽  
Author(s):  
François Guéritaud ◽  
Fanny Kassel
Keyword(s):  
Hyperbolic Manifolds ◽  
Geometrically Finite

scholarly journals Power spectrum of the geodesic flow on hyperbolic manifolds

2015 ◽  
Vol 8 (4) ◽  
pp. 923-1000 ◽  
Author(s):  
Semyon Dyatlov ◽  
Frédéric Faure ◽  
Colin Guillarmou
Keyword(s):  
Power Spectrum ◽  
Geodesic Flow ◽  
Hyperbolic Manifolds

Chains that realize the Gromov invariant of hyperbolic manifolds

1997 ◽  
Vol 17 (3) ◽  
pp. 643-648 ◽  
Author(s):  
DOUGLAS JUNGREIS
Keyword(s):  
Hyperbolic Manifold ◽  
Homology Class ◽  
Hyperbolic Manifolds ◽  
Uniform Measure ◽  
Geodesic Boundary ◽  
Maximal Volume ◽  
Totally Geodesic ◽  
Gromov Norm

For any closed hyperbolic manifold of dimension $n \geq 3$, suppose a sequence of $n$-cycles representing the fundamental homology class have norms converging to the Gromov invariant. We show that this sequence must converge to the uniform measure on the space of maximal-volume ideal simplices. As a corollary, we show that for a hyperbolic $n$-manifold $L$ ($n \geq 3$) with totally-geodesic boundary, the Gromov norm of ($L,\partial L$) is strictly greater than the volume of $L$ divided by the maximal volume of an ideal $n$-simplex.

scholarly journals Ruelle and Quantum Resonances for Open Hyperbolic Manifolds

2018 ◽  
Vol 2020 (5) ◽  
pp. 1445-1480
Author(s):  
Charles Hadfield
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifolds ◽  
Quantum Resonances ◽  
Classical Quantum ◽  
Convex Cocompact ◽  
Quantum Correspondence

Abstract We establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds between the spectra of the geodesic flow and the Laplacian acting on natural tensor bundles. This extends previous work detailing the correspondence for cocompact quotients.

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