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Scott A. Wolpert - One of the best experts on this subject based on the ideXlab platform.
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Topological dynamics of the Weil-Petersson Geodesic Flow
Advances in Mathematics, 2010Co-Authors: Mark Pollicott, Howard M. Weiss, Scott A. WolpertAbstract:We prove topological transitivity for the Weil–Petersson Geodesic Flow for real two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that combines the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil–Petersson Geodesic Flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed Geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite.
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Topological dynamics of the Weil-Petersson Geodesic Flow
arXiv: Dynamical Systems, 2007Co-Authors: Mark Pollicott, Howard M. Weiss, Scott A. WolpertAbstract:We prove topological transitivity for the Weil Petersson Geodesic Flow for two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that exploits the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil Petersson Geodesic Flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed Geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite.
Amie Wilkinson - One of the best experts on this subject based on the ideXlab platform.
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rates of mixing for the weil petersson Geodesic Flow exponential mixing in exceptional moduli spaces
Geometric and Functional Analysis, 2017Co-Authors: Keith Burns, Howard Masur, Carlos Matheus, Amie WilkinsonAbstract:We establish exponential mixing for the Geodesic Flow \({\varphi_t\colon T^1S\to T^1S}\) of an incomplete, negatively curved surface S with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil–Petersson Flows for the moduli spaces \({\mathcal{M}_{1,1}}\) and \({\mathcal{M}_{0,4}}\) are exponentially mixing, in sharp contrast to the Flows for \({\mathcal{M}_{g,n}}\) with \({3g-3+n > 1}\), which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic Flows with singularities, based on changing the Riemannian metric on the phase space T 1 S and rescaling the Flow \({\varphi_t}\).
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ergodicity of the weil petersson Geodesic Flow
2017Co-Authors: Keith Burns, Howard Masur, Amie WilkinsonAbstract:We describe the proof that the Geodesic Flow for the Weil–Petersson metric on the moduli space of Riemann surfaces is ergodic and in fact Bernoulli. Other chapters in this volume complement this summary by describing in depth the needed implementation of the Hopf argument and some of the pertinent aspects of moduli spaces of Riemann surfaces (and Teichmuller theory).
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The Weil-Petersson Geodesic Flow is ergodic
Annals of Mathematics, 2012Co-Authors: Keith Burns, Howard Masur, Amie WilkinsonAbstract:We prove that the Geodesic Flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (and in fact Bernoulli) and has finite, positive metric entropy.
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Stable ergodicity of the time-one map of a Geodesic Flow
Ergodic Theory and Dynamical Systems, 1998Co-Authors: Amie WilkinsonAbstract:We prove that the time-one map of the Geodesic Flow for a closed, negatively curved surface is stably ergodic.
Mark Pollicott - One of the best experts on this subject based on the ideXlab platform.
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Topological dynamics of the Weil-Petersson Geodesic Flow
Advances in Mathematics, 2010Co-Authors: Mark Pollicott, Howard M. Weiss, Scott A. WolpertAbstract:We prove topological transitivity for the Weil–Petersson Geodesic Flow for real two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that combines the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil–Petersson Geodesic Flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed Geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite.
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Ergodicity of the Geodesic Flow on non-complete negatively curved surfaces
Asian Journal of Mathematics, 2009Co-Authors: Mark Pollicott, Howard M. WeissAbstract:We study the dynamics of the Geodesic Flow for a class of non-complete Riemannian metrics on a negatively curved surface M . These finite area surfaces are composed of finitely many “singular” surfaces of revolutions of the form y = x, r > 1 for 0 ≤ x ≤ 1 (thin pieces), together with connecting surfaces of bounded negative curvature (thick pieces). The curvature at every point is negative and bounded away from zero, but is unbounded in the “cusps.” The Geodesic Flow is non-complete because Geodesics corresponding to singular unit tangent vectors pointing into the origin hit the cusps in finite time and then cease to be defined. Such singular unit tangent vectors are dense in the unit tangent bundle. Our main result is ergodicity of the Geodesic Flow (Theorem 5.1). It immediately follows that these metrics have dense Geodesics in the unit tangent bundle SM . We also prove that the closed Geodesics are dense in the unit tangent bundle (Theorem 6.1) The motivation for considering these metrics arises from studying the Geodesic Flow for the Weil-Petersson (WP) metric on two dimensional moduli spaces of Riemann surfaces., e.g., the moduli space for the once punctured torus. In several fundamental ways the Geodesic Flow we consider provides a good model for the WP Geodesic Flow. For example, Wolpert showed that the WP metric is also non-complete, has finite area, has negative curvature bounded away from zero, and is unbounded in the ”cusps.” Also, various authors have shown that the cusps can be approximated by singular surface of revolutions. At present, there is only a single technical obstruction (additional estimates on the derivatives of the GF in the cusp) that prevent us from applying this general method to prove ergodicity of the WP Geodesic Flow. See Section 7. The geometric properties of both types of Flows imply that the Geodesic Flow is a uniformly hyperbolic dynamical system with singularities. The non-completeness causes pathologies in the stable and unstable manifolds, as for many billiard Flows. For example, stable and unstable manifolds at a point, if they exist, may intersect a cusp point, and thus have only finite length. One needs extensions of “Pesin theory” to systems with singularities to study these Geodesic Flows. Another challenging aspect is that these surfaces may be simply connected, making the use of any of the traditional arguments involving boundaries of the covering spaces inapplicable. Studying the dynamics requires the development of new approches. Our strategy to prove ergodicity of the Geodesic Flow is to first establish non-uniform hyperbolicity, i.e., the Lyapunov exponents are non-zero at almost every point. We then prove that there
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Topological dynamics of the Weil-Petersson Geodesic Flow
arXiv: Dynamical Systems, 2007Co-Authors: Mark Pollicott, Howard M. Weiss, Scott A. WolpertAbstract:We prove topological transitivity for the Weil Petersson Geodesic Flow for two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that exploits the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil Petersson Geodesic Flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed Geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite.
Howard Masur - One of the best experts on this subject based on the ideXlab platform.
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Divergent on average directions of Teichmuller Geodesic Flow
arXiv: Dynamical Systems, 2018Co-Authors: Paul Apisa, Howard MasurAbstract:The set of directions from a quadratic differential that diverge on average under Teichmuller Geodesic Flow has Hausdorff dimension exactly equal to one-half.
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rates of mixing for the weil petersson Geodesic Flow exponential mixing in exceptional moduli spaces
Geometric and Functional Analysis, 2017Co-Authors: Keith Burns, Howard Masur, Carlos Matheus, Amie WilkinsonAbstract:We establish exponential mixing for the Geodesic Flow \({\varphi_t\colon T^1S\to T^1S}\) of an incomplete, negatively curved surface S with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil–Petersson Flows for the moduli spaces \({\mathcal{M}_{1,1}}\) and \({\mathcal{M}_{0,4}}\) are exponentially mixing, in sharp contrast to the Flows for \({\mathcal{M}_{g,n}}\) with \({3g-3+n > 1}\), which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic Flows with singularities, based on changing the Riemannian metric on the phase space T 1 S and rescaling the Flow \({\varphi_t}\).
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ergodicity of the weil petersson Geodesic Flow
2017Co-Authors: Keith Burns, Howard Masur, Amie WilkinsonAbstract:We describe the proof that the Geodesic Flow for the Weil–Petersson metric on the moduli space of Riemann surfaces is ergodic and in fact Bernoulli. Other chapters in this volume complement this summary by describing in depth the needed implementation of the Hopf argument and some of the pertinent aspects of moduli spaces of Riemann surfaces (and Teichmuller theory).
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The Weil-Petersson Geodesic Flow is ergodic
Annals of Mathematics, 2012Co-Authors: Keith Burns, Howard Masur, Amie WilkinsonAbstract:We prove that the Geodesic Flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (and in fact Bernoulli) and has finite, positive metric entropy.
Silvere Bonnabel - One of the best experts on this subject based on the ideXlab platform.
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a simple intrinsic reduced observer for Geodesic Flow
IEEE Transactions on Automatic Control, 2010Co-Authors: Silvere BonnabelAbstract:Aghannan and Rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. The (position and velocity) observer is based on the Riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic. They proved local convergence. When the system is conservative, we propose an intrinsic reduced order (velocity) observer based on the Jacobi metric, which can be initialized such that it converges exponentially for any initial true velocity. For non-conservative systems the observer can be used as a complement to the one of Aghannan and Rouchon. More generally the reduced observer provides velocity estimation for Geodesic Flow with position measurements. Thus it can be (formally) used as a fluid Flow soft sensor in the case of a perfect incompressible fluid. When the curvature is negative in all planes the Geodesic Flow is sensitive to initial conditions. Surprisingly in this case we have global exponential convergence and the more unstable the Flow is, faster is the convergence.
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A simple intrinsic reduced-observer for Geodesic Flow
arXiv: Optimization and Control, 2008Co-Authors: Silvere BonnabelAbstract:Aghannan and Rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. The observer is based on the Riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic. They proved local convergence. When the system is conservative, we propose a globally convergent intrinsic reduced-observer based on the Jacobi metric. For non-conservative systems the observer can be used as a complement to the one of Aghannan and Rouchon. More generally the reduced-observer provides velocity estimation for Geodesic Flow with position measurements. Thus it can be (formally) used as a fluid Flow soft sensor in the case of a perfect incompressible fluid. When the curvature is negative in all planes the Geodesic Flow is sensitive to initial conditions. Surprisingly this instability yields faster convergence.