scholarly journals Escape of mass and entropy for geodesic flows

2017 ◽  
Vol 39 (2) ◽  
pp. 446-473
Author(s):  
FELIPE RIQUELME ◽  
ANIBAL VELOZO
Keyword(s):  
Phase Transition ◽  
Geodesic Flow ◽  
Upper Semicontinuity ◽  
Geodesic Flows ◽  
Parabolic Subgroups ◽  
Hölder Continuous ◽  
Upper Semicontinuous ◽  
Negatively Curved ◽  
Geometrically Finite ◽  
Loss Of Mass

In this paper, we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure-theoretical entropy is upper semicontinuous when there is no loss of mass. In the case where mass is lost, the critical exponents of parabolic subgroups of the fundamental group have a significant meaning. More precisely, the failure of upper-semicontinuity of the entropy is determined by the maximal parabolic critical exponent. We also study the pressure of positive Hölder-continuous potentials going to zero through the cusps. We prove that the pressure map $t\mapsto P(tF)$ is differentiable until it undergoes a phase transition, after which it becomes constant. This description allows us, in particular, to compute the entropy of the geodesic flow at infinity.

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

Expansive geodesic flows on surfaces

1993 ◽  
Vol 13 (1) ◽  
pp. 153-165 ◽  
Author(s):  
Miguel Paternain
Keyword(s):  
Geodesic Flow ◽  
Compact Surface ◽  
Conjugate Points ◽  
Riemannian Surface ◽  
Geodesic Flows ◽  
Flows On Surfaces

AbstractWe prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

scholarly journals A generalised quasi-variational inequality without upper semicontinuity

1996 ◽  
Vol 54 (2) ◽  
pp. 247-254 ◽  
Author(s):  
Paolo Cubiotti ◽  
Xian-Zhi Yuan
Keyword(s):  
Variational Inequality ◽  
Convex Subset ◽  
Existence Result ◽  
Upper Semicontinuity ◽  
Nonempty Closed Convex Subset ◽  
Positive Answer ◽  
Upper Semicontinuous ◽  
Quasi Variational Inequality ◽  
General Existence ◽  
Image Position

In this note we deal with the following problem: given a nonempty closed convex subset X of Rn and two multifunctions Γ : X → 2X and , to find ( such thatWe prove a very general existence result where neither Γ nor Φ are assumed to be upper semicontinuous. In particular, our result give a positive answer to an open problem raised by the first author recently.

scholarly journals Gibbs measures for foliated bundles with negatively curved leaves

2016 ◽  
Vol 38 (4) ◽  
pp. 1238-1288 ◽  
Author(s):  
SÉBASTIEN ALVAREZ
Keyword(s):  
Invariant Measure ◽  
Gibbs Measure ◽  
Geodesic Flow ◽  
Gibbs Measures ◽  
Holonomy Group ◽  
Bijective Correspondence ◽  
Weighted Averages ◽  
Negatively Curved ◽  
Harmonic Measures ◽  
Foliated Bundle

In this paper we develop a notion of Gibbs measure for the geodesic flow tangent to a foliated bundle over a compact negatively curved base. We also develop a notion of$F$-harmonic measure and prove that there exists a natural bijective correspondence between these two concepts. For projective foliated bundles with$\mathbb{C}\mathbb{P}^{1}$-fibers without transverse invariant measure, we show the uniqueness of these measures for any Hölder potential on the base. In that case we also prove that$F$-harmonic measures are realized as weighted limits of large balls tangent to the leaves and that their conditional measures on the fibers are limits of weighted averages on the orbits of the holonomy group.

scholarly journals Approximate solutions of cohomological equations associated with some Anosov flows

1990 ◽  
Vol 10 (2) ◽  
pp. 367-379 ◽  
Author(s):  
Svetlana Katok
Keyword(s):  
Approximate Solutions ◽  
Geodesic Flows ◽  
Curved Surfaces ◽  
Anosov Flow ◽  
3 Dimensional ◽  
Negatively Curved ◽  
Anosov Flows ◽  
Higher Differentiability ◽  
Special Case ◽  
Cohomological Equations

AbstractThe Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

scholarly journals Dynamics of the geodesic flow of a foliation

1988 ◽  
Vol 8 (4) ◽  
pp. 637-650 ◽  
Author(s):  
Paweł G. Walczak
Keyword(s):  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Geodesic Flow ◽  
Smooth Measure ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Foliated Riemannian Manifold

AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

Finite-dimensional coagulation-fragmentation dynamics

2018 ◽  
Vol 28 (05) ◽  
pp. 851-868
Author(s):  
Jack Carr ◽  
Matab Alghamdi ◽  
Dugald B. Duncan
Keyword(s):  
Phase Transition ◽  
Asymptotic Behaviour ◽  
Numerical Approximation ◽  
Finite System ◽  
Numerical Results ◽  
Centre Manifold ◽  
Finite Dimensional ◽  
Time Period ◽  
Fragmentation Dynamics ◽  
Loss Of Mass

We examine a finite-dimensional truncation of the discrete coagulation-fragmentation equations that is designed to allow mass to escape from the system into clusters larger than those in the truncated problem. The aim is to model within a finite system the process of gelation, which is a type of phase transition observed in aerosols, colloids, etc. The main result is a centre manifold calculation that gives the asymptotic behaviour of the truncated model as time [Formula: see text]. Detailed numerical results show that truncated system solutions are often very close to this centre manifold, and the range of validity of the truncated system as a model of the full infinite problem is explored for systems with and without gelation. The latter cases are mass conserving, and we provide an estimate using quantities from the centre manifold calculations of the time period and the truncated system can be used for before loss of mass which is apparent. We also include some observations on how numerical approximation can be made more reliable and efficient.

Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows

1997 ◽  
Vol 17 (5) ◽  
pp. 1043-1059 ◽  
Author(s):  
KEITH BURNS ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Positive Measure ◽  
Geodesic Flows ◽  
Open Set ◽  
Negative Questions

Let $M$ be a compact $C^{\infty}$ Riemannian manifold. Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number of geodesic segments joining $p$ and $q$ with length $\leq T$. Mañé showed in [7] that \[ \lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq = h_{\rm top}, \] where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of $M$.In this paper we exhibit an open set of metrics on the two-sphere for which \[ \limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top}, \] for a positive measure set of $(p,q)\in M\times M$. This answers in the negative questions raised by Mañé in [7].

scholarly journals Harmonic measures and the foliated geodesic flow for foliations with negatively curved leaves

2014 ◽  
Vol 36 (2) ◽  
pp. 355-374 ◽  
Author(s):  
SÉBASTIEN ALVAREZ
Keyword(s):  
Gibbs Measure ◽  
Geodesic Flow ◽  
Bijective Correspondence ◽  
Negatively Curved ◽  
Harmonic Measures

In this paper we define a notion of Gibbs measure for the geodesic flow tangent to a foliation with negatively curved leaves and associated to a particular potential $H$. We prove that there is a canonical bijective correspondence between these measures and Garnett’s harmonic measures.

scholarly journals Coding of geodesics and Lorenz-like templates for some geodesic flows

2016 ◽  
Vol 38 (3) ◽  
pp. 940-960
Author(s):  
PIERRE DEHORNOY ◽  
TALI PINSKY
Keyword(s):  
Periodic Orbits ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Hyperbolic Metric ◽  
Geodesic Flows ◽  
Unit Tangent Bundle

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

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