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Topic: Geodesic flow

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	Geodesic - Wikipedia, the free encyclopedia
		In physics, geodesics describe the motion of point particles; in particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all described by geodesics in the theory of general relativity.
		The geodesic flow defines a family of curves on the tangent manifold.
		Formally, the tangent bundle to the tangent manifold is known as a jet bundle; thus the geodesic spray is a vector field in the (first) jet bundle of the manifold.
	en.wikipedia.org /wiki/Geodesic_flow (1074 words)

	Flow - Wikipedia, the free encyclopedia
		In psychology, flow is the feeling of complete and energized focus in an activity, with a high level of enjoyment and fulfillment as described by Mihaly Csikszentmihalyi.
		In economics and accounting, a flow is the total value of sales or purchases in an accounting period, usually a quarter or a year.
		The notion of network flow is formalized in a branch of mathematics known as graph theory.
	en.wikipedia.org /wiki/Flow (305 words)

	Exponential map
		In differential geometry, the exponential map is the map from a subset of the tangent space of a Riemannian manifold M to M itself.
		The geodesic flow is the corresponding flow on the tangent bundle TM of M.
		The name comes from the fact that it coincides with exponentiation of matrices in the case of certain metrics on Lie groups, when one is using a matrix representation of the group, and its Lie algebra as tangent space at the identity.
	www.teachtime.com /en/wikipedia/e/ex/exponential_map.html (138 words)

	Ergodicity Of Harmonic Invariant Measures For The Geodesic Flow On Hyperbolic Spaces - Kaimanovich (ResearchIndex)
		Abstract: this paper is to prove an analogue of the Hopf dichotomy for the harmonic measure in this generality: either the quotient Markov operator is recurrent and the geodesic flow is ergodic with respect to the harmonic invariant measure, or the quotient operator is transient and the geodesic flow is completely dissipative.
		Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. reine angew.
		8 Ergodic theory and the geodesic flow on surfaces of constant..
	citeseer.ist.psu.edu /kaimanovich94ergodicity.html (888 words)

	Geodesic (Site not responding. Last check: 2007-10-21)
		In intuitive terms, an elastic band stretched along a path that is not geodesic would contract its length for energy reasons to a nearby shorter path — this though only serves to explain that a geodesic is a local minimum for length.
		Geodesics play an important role in the theory of general relativity, where they are the world lines of a particle free from all external force; see the main article geodesic (general relativity) for details.
		The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term.
	www.worldhistory.com /wiki/G/Geodesic.htm (1095 words)

	Schedule for 2001/02 (Site not responding. Last check: 2007-10-21)
		Geodesic flow on negatively curved manifold and the rigidity problem.I. Tsarev (Krasnoyarsk).
		Geodesic flow on negatively curved manifold and the rigidity problem.IV.
		Geodesic flow on negatively curved manifold and the rigidity problem.V. Yu.G. Nikonorov (Rubtsovsk).
	www.math.nsc.ru /~taimanov/schedules/2001-en.htm (300 words)

	Literature
		A well-known, classical result of Hadamard [1] is that the geodesic flow on a surface of genus greater than 1 is non-integrable - regardless of the metric, so long as it is analytic.
		The methods that are used to prove these theorems are very similar to the techniques that are used to show that the fundamental group of a compact manifold with everywhere negative sectional curvature has exponential growth and has only cyclic abelian subgroups [6].
		The proof of the second and third conclusions relies on results due to Gromov [7, 8] and Yomdin [16] which place lower bounds on the entropy of a cogeodesic flow in terms of the rate of growth of the betti numbers of the loop space.
	www.mast.queensu.ca /~lbutler/thesisproposal/node4.html (1006 words)

	My publications (as on June 2005)
		Metric with ergodic geodesic flow is completely determined by unparameterized geodesics, (with P. Topalov), ERA-AMS, 6(2000).
		Geodesic equivalence of metrics as a particular case of integrability of geodesic flows, (with P. Topalov), Theor.
		Geodesic equivalence of metrics as a particular case of integrability of geodesic flows, (with P. Topalov), MPIM Preprint Series, 1999(47).
	home.mathematik.uni-freiburg.de /matveev/Forschung/publications.html (845 words)

	Hamiltonian mechanics -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The (additional info and facts about integral curve) integral curves of the vector field are a one-parameter family of transformations of the manifold; the parameter of the curves is commonly called the time.
		In particular, the (additional info and facts about Hamiltonian flow) Hamiltonian flow in this case is the same thing as the (additional info and facts about geodesic flow) geodesic flow.
		The existence of sub-Riemannian geodesics is given by the Chow-Rashevskii theorem.
	www.absoluteastronomy.com /encyclopedia/h/ha/hamiltonian_mechanics.htm (1682 words)

	Victor Donnay (Bryn Mawr) (Site not responding. Last check: 2007-10-21)
		It is known that no metric on surfaces of genus 0 or 1 can have an Anosov geodesic flow, while for surfaces of genus 2 or higher there exist metrics of negative curvature that produce Anosov flows, but these negatively curved metrics can not be isometrically embedded.
		Our construction produces surfaces with Anosov geodesic flow for all sufficiently large genus, but does not produce any explicit bounds on the genus.
		Pugh and builds on their earlier work of constructing embedded surfaces with Bernoulli geodesic flow using the finite horizon cap construction.
	www.math.psu.edu /dynsys/DW2001/abstracts/node6.html (129 words)

	Preprints of Leo Butler (Site not responding. Last check: 2007-10-21)
		As a second consequence of this construction, the geodesic flow of a surface of constant negative curvative is embedded as a subsystem of an integrable geodesic flow.
		Then there is a riemannian $9$-manifold $(\Sigma,{\bf k})$ such that $\Sigma$ is homotopy equivalent to $M$, the geodesic flow of ${\bf k}$ is completely integrable and the geodesic flow of ${\bf h}$ is a subsystem of ${\bf k}$'s.
		In this paper, the degeneracy of the Poisson tensor on the dual algebra is shown to be the source of the large number of commuting first integrals, and additional examples of integrable geodesic flows are constructed.
	www.mast.queensu.ca /~lbutler/preprint.html (1228 words)

	Abstract (Site not responding. Last check: 2007-10-21)
		The geodesic flow on a simply connected manifold M is the R-action by translations on its unit tangent bundle SM.
		In a joint work with Monod and Shalom we construct a homological version of geodesic flow for an arbitrary hyperbolic group, called the ideal bicombing.
		I would like to discuss this construction and also to present a metric version of geodesic flow that generalizes and combines the results known so far.
	www.math.columbia.edu /~ikofman/abstracts/mineyev.html (135 words)

	Stony Brook Math Calendar
		The geodesic flow on a manifold M is the R-action by translations on its unit tangent bundle SM.
		Gromov gave an outline of a geodesic flow construction for a general hyperbolic group, and several other results by a number of authors appeared after that.
		This geodesic flow is a part of a much more general concept which should be called symmetric join, and which can be defined for any topological space X. If X is a metric space, there is a canonical way to define a metric on the symmetric join of X. March 02, 2004
	www.math.sunysb.edu /~calendar/scott.php?LocationID=8&Date=2004-01-26 (760 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		For the geodesic flow on a negatively curved manifold, it is conjectured to hold as soon as the covering manifold is non elementary, and known to be true, for example, if there is a cusp, or if the non-wandering set of the flow on the cover is the whole manifold.
		For a geodesic flow on a non-elementary negatively curved manifold, this was proven by F. Dalbo.
		So, if the flow is positively transitive and admits a point with a dense leaf, then there is a (dense set of) positively transitive points with dense strong stable leaf.
	name.math.univ-rennes1.fr /yves.coudene/commentHN.html (2175 words)

	Report of Project 98-1 (Site not responding. Last check: 2007-10-21)
		The results about the linearly-integrable and quadratic-integrable geodesic flows were obtained, up to most natiral equivalences, namely: up to isometry, orbital equivalence, geodesic equivalence and Liouville equivalence.
		Bolsinov A.V., Taimanov I.A. "An example of an integrable geodesic flow with positive topological entropy", Proceedings of the International Conference dedicated to the 80th birthday of V.A.Rochlin, Topology and Dynamics, St,Petersburg, August 19-25, 1999, p.14.
		On the classification of Morse-Smale flows on two-dimensional manifolds.
	liapunov.inria.msu.ru /reports/1999_2000/98-01.html (3142 words)

	[No title]
		\begin{abstract} The Ruelle zeta-function of the geodesic flow on the sphere bundle $S(X)$ of an even-dimensional compact locally symmetric space $X$ of rank $1$ is a meromorphic function in the complex plane that satisfies a functional equation relating its values in $s$ and $-s$.
		Let $\Phi_t$ be the geodesic flow on the unit sphere bundle $S(X)$ of the space $(X, g)$.
		Note that for each (unoriented) closed geodesic $c$ in $X$ there are two lifts of $c$ as periodic orbits of $\Phi_t$ in correspondence with the two possibilities to orient $c$.
	www.ams.org /journals/bull/pre-1996-data/199501/199501009.tex.html (3227 words)

	Victor Donnay’s research area is the ergodic properties of smooth dynamical systems, particularly geodesic flow on ... (Site not responding. Last check: 2007-10-21)
		Victor Donnay’s research area is the ergodic properties of smooth dynamical systems, particularly geodesic flow on surfaces and billiard systems.
		He has found new examples of surfaces, including surfaces that are embedded in three dimensional Euclidean space, for which the geodesic flow is chaotic (ergodic).
		Geodesic flow on the two-sphere, Part I: Positive measure entropy, Ergod.
	serendip.brynmawr.edu /local/keck/proposals/donnay.html (347 words)

	[No title]
		For such a manifold, the set of closed geodesics is dense in the set of all geodesics.
		Nevertheless, only nonpositivity of the curvature and Anosov type of the geodesic flow are used in the proof.
		In the case of an $ n $-dimensional Anosov manifold $ M $, the stable and unstable distributions are defined on the manifold $ \Omega M $ of unit tangent vectors.
	www.ma.utexas.edu /mp_arc/papers/98-412 (4479 words)

	abstract_flow (Site not responding. Last check: 2007-10-21)
		This evolution is described by the geodesic flow and helps us to understand the geometry of surfaces.
		In this paper we compute the evolution of distance circles on polyhedral surfaces and develop a method to visualize the set of circles, their overlapping, branching, and their temporal evolution simultaneously.
		We consider the evolution as an interfering wave on the surface, and extend isometric texture maps to efficiently handle the branching and overlapping of the wave.
	www.math.tu-berlin.de /~schmies/homepage/article/abstract_flow.htm (114 words)

	Abstract by Dr. Alex Furman (Site not responding. Last check: 2007-10-21)
		(Joint work with Benjamin Weiss) The geodesic flow on a compact Riemann surface M can be identified with the action the diagonal \{g_t = diag(e^t,e^{-t})\} of G = SL_2({\bf R}) on the homogeneous space G/\Gamma, where \Gamma is isomorphic with the fundamental group of M. Geodesic flows are known to be Bernoullian.
		We consider the following {\it generalized geodesic flow}: let G = SL_2({\bf R) act {\it ergodically} (rather than transitively) on a probability space (X,m), then consider the ergodic properties of the flow (X,m,\{g_t\}).
		It turns out, that although the flow (X,m,\{g_t\}) is always a K-flow, there exists a natural class of examples, in which it is not Bernoullian.
	www.math.uiuc.edu /Bulletin/Abstracts/February/feb14-97analysis.html (151 words)

	UBC Mathematics Department - Colloquium (Site not responding. Last check: 2007-10-21)
		We study fractal-like geometric objects by means of the flow defined by zooming toward a point of an ambient Euclidean space.
		This ``scenery flow" provides an analogue for the geodesic flow associated to a Kleinian group.
		This observation builds a bridge between fractal geometry and the probability theory of recurrent events, suggesting on the one hand new theorems for the Fuchsian case and on the other a new interpretation of some results on countable state Markov chains due to Feller and Chung-Erdös.
	www.math.ubc.ca /Dept/Events/colloquia/DSSeminarSept15.html (141 words)

	mp_arc 98-693 (Site not responding. Last check: 2007-10-21)
		Spheres with positive curvature and nearly dense orbits for the geodesic flow (643K, Postscript) Oct 30, 98
		For any $\ep > 0$, we construct an explicit smooth Riemannian metric on the sphere $S^n, n \geq 3$, that is within $\ep$ of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is $\ep$-dense in the unit tangent bundle.
		Moreover, for any $\ep > 0$, we construct a smooth Riemannian metric on $S^n, n \geq 3$, that is within $\ep$ of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesic flow has Liouville measure less than $\ep$.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=98-693 (114 words)

	[No title]
		The map (3) is essentially given by the geodesic flow and discussed in detail in Section 4.
		Also we extend the geodesic foliation (by taking the prod* uct with the trivial 0-dimensional foliation of B) to a foliation Fw of SHM~ x B. ow Ew is defined similar to Egeo as foliated control with a certain carefully chos* *en decay speed S (defined in Subsection 6.3) with respect to the foliation Fw.
		First recall that for a cell e in a single flow cel* *l structure L we have the constants ffle and Ce appearing in Definition 6.1 and Remark 6.2.
	hopf.math.purdue.edu /Bartels-Reich/isoIIhopf.txt (13817 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The depth-averaged 3D Euler equations for shallow water flow are well approximated by 2D equations of geodesic motion for a certain Sobolev norm.
		The initial value problem for these equations produces filamentary soliton-like solutions that are measure-valued, that is, they are supported on curves that evolve in the plane.
		The existence of these measure-valued solutions of geodesic flow is guaranteed -- with any Sobolev norm, and in any number of spatial dimensions -- because the solution ansatz is a momentum map for the action of diffeomorphisms on the measure-valued support set of the solutions.
	www.ipam.ucla.edu /abstract.aspx?tid=3837 (234 words)

	Math 557B Lecture Notes (Site not responding. Last check: 2007-10-21)
		conjugacy to the (flow of the) equation x'=cw, where c= is the mean value of a over the torus (proof deferred to the next lecture).
		For fixed value of energy, all trajectories of each of the types are equivalent to a geodesic flow on a surface of constant negative curvature.
		Theorem: The geodesic flow on a compact surface of constant negative curvature is ergodic.
	alamos.math.arizona.edu /~rychlik/557-dir/557B-notes.html (796 words)

	Research (Site not responding. Last check: 2007-10-21)
		Whereas for real matrices the problem can be reduced to the dynamics of geodesic and horocycle flows on surfaces, the generalizations come from a study of frame flows, and the associated strong stable foliations.
		A nice solution lies in considering the associated geodesic flow, then closed orbits for the flow correspond to closed geodesics and we have a dynamical problem which we can solve - by using ideas from number theory.
		A frame flow is a natural generalization of a geodesic flow on a negatively curved manifold..
	www.ma.man.ac.uk /~mp/research.html (1119 words)

	IRMA Strasbourg - Publication 1999 (Site not responding. Last check: 2007-10-21)
		Symbolic coding for the geodesic flow associated to a word hyperbolic group.
		Gromov has constructed a compact space $\overline{G}=\overline{G}(\Gamma)$ equipped with a flow which is defined up to orbit-equivalence and which is called the geodesic flow of $\Gamma$.
		In the special case where $\Gamma$ is the fundamental group of a Riemannian manifold of negative sectional curvature, $\overline{G}$ is the unit tangent bundle of the manifold equipped with the usual geodesic flow.
	www-irma.u-strasbg.fr /irma/publications/1999/99018.shtml (167 words)

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