Kiho Park, Tianyu Wang
We study multifractal spectra of the geodesic flows on rank 1 surfaces without focal points. We compute the entropy of the level sets for the Lyapunov exponents and estimate its Hausdorff dimension from below. In doing so, we employ and generalize results of Burns and Gelfert.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
If $(M,g)$ is a smooth compact rank $1$ Riemannian manifold without focal points, it is shown that the measure $μ_{\max}$ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $μ_{\max}$ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $SM$ with respect to $μ_{\max}$ is Bernoulli is acquired provided $M$ is a compact surface with genus greater than one and no focal points.
We prove that for closed rank 1 manifolds without focal points the equilibrium states are unique for Hölder potentials satisfying the pressure gap condition. In addition, we provide a criterion for a continuous potential to satisfy the pressure gap condition. Moreover, we derive several ergodic properties of the unique equilibrium states including the equidistribution and the K-property.
In this article, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let $M$ be a smooth connected and closed surface equipped with a $C^\infty$ Riemannian metric $g$, whose genus $\mathfrak{g} \geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tangent bundle of $M$ is ergodic with respect to the Liouville measure, under the assumption that the set of points on $M$ with negative curvature has at most finitely many connected components.
In this article, we consider a closed rank one Riemannian manifold $M$ without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates: \[\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\] where $h$ is the topological entropy of the geodesic flow. In the appendix, we also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.
In this article, we consider the geodesic flow on a compact rank $1$ Riemannian manifold $M$ without focal points, whose universal cover is denoted by $X$. On the ideal boundary $X(\infty)$ of $X$, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on $M$ has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in $X$ and the growth rate of the number of closed geodesics on $M$. These r…
In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.
Multifractal analysis of homological growth rates for hyperbolic surfaces
该研究对双曲曲面上定向测地线同调增长率开展多重分形分析，给出了由富克斯群广义庞加莱指数表示的水平集豪斯多夫维数公式，并借助符号动力学等方法证明了维数谱的解析性。
A self-focal point of a Riemannian manifold $(M,g)$ is a point $p$ so that every geodesic starting from $p$ returns to $p$ at some positive time. It is called a pole if all geodesics through $p$ are closed, and a non-polar self-focal point if all geodesics loop back but not all are smoothly closed. Umbilic points of two dimensional tri-axial ellipsoids are non-polar self-focal points. Little is known about existence of self-focal points for Riemannian manifolds of dimension $\geq 3$. We prove that ellipsoids of dimension $\geq 3$ with at least 4 distinct axes have no self-focal points. Certain…
Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time-parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.
A multifractal analysis for escaping trajectories on free groups
本文研究自由群扩张下有限型子移位的逃逸轨迹，给出三类逃逸点集u-维数的共同正下界及对称条件下的等式，并应用于Schottky曲面测地流，证明存在维数间隙。
Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces
本文研究了双曲面上多尖点旋绕过程的多标度分析，建立了其Hausdorff维度谱的条件变分原理，并证明了该维度谱的解析性。
Density of expansivity for geodesic flows of compact higher genus surfaces without conjugate points
本文证明了在亏格大于1且无共轭点的紧致连通光滑曲面中，无带状区域的测地集合在单位切丛中构成稠密轨道集，推广了无焦点条件下的已知结论。
Geometric conditions to obtain Anosov geodesic flow in non-compact manifolds
本文研究完备非紧无焦点流形上测地流成为 Anosov 型的几何判据，证明沿测地线的切平面截面曲率平均一致为负即可保证该性质，并据此构造非正曲率非紧例子。
Rigidity of Lyapunov Exponents for Geodesic Flows
本文研究测地流沿周期轨道的 Lyapunov 指数与几何结构的刚性关系，证明无焦点曲面及 Anosov 测地流情形下常值指数蕴含平坦环面或常负曲率，并推广至有限体积情形。
In this paper we prove that if the geodesic flow of a {compact or non-compact} complete manifold without conjugate points is of the Anosov type, then the average of the integral of the sectional curvature along the geodesic is negative and away from zero from a uniform time. Moreover, in dimension two, if the manifold has no focal points, then this condition is sufficient to obtain that the geodesic flow is of Anosov type. This sufficient condition will also be used to construct new examples of non-compact surfaces whose geodesic flow is of the Anosov type.
Let M be a possibly non compact smooth manifold. We study genericity in the C^k-topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics satisfying suitable endpoints conditions. This extends results in the literature for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of MxM that satisfies an admissibility condition. Immediate consequences are generic non conjugacy betwee…
Curvature Rigidity Through Level Sets of Lyapunov Exponents in Geodesic Flows
本文通过不变测度研究Lyapunov指数水平集，证明零Lyapunov指数水平集全测度蕴含流形平坦、水平集全测度蕴含截面曲率常数，并给出无共轭点流形上阻碍光滑共轭的新判据，统一证明了已有刚性结果。
We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass phenomenon and the measure theoretic entropy on finite volume nonpositively curved manifolds satisfying the Visibility axiom. We provide a general criterion for the same relation to hold between the escape of mass and the measure theoretic entropy. This gives a criterion for the existence of measures of maximal entropy for the geodesic flow on some nonpositively c…
An upper bound for the mean Lorentz force of magnetic flows without conjugate points in terms of the geodesic curvature of horocycles
本文研究亏格至少二的闭黎曼曲面上无共轭点的磁流，证明在 Mañé 临界值之上，洛伦兹力积分受水平圆测地曲率约束，并由此推出基于面积和高斯曲率的显式上界。
Isolated Flat Points and $C^2$-Robustness of the No Focal Points Property
本文证明紧致黎曼曲面上非正曲率且零曲率集有限的度量属于无焦点度量集合的 $C^2$-内部，并构造了位于负曲率边界但仍在无焦点区域内的共形度量。
A Metric Entropy Formula for Magnetic Flows without Conjugate Points
本文利用磁系统的新术语，将测地流理论中的Freire-Mañé定理推广到磁流，并证明无共轭点的能量层上度量熵为正或磁曲率恒为零。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。