Gromov-hyperbolicity and transitivity of geodesic flows in n-dimensional Finsler manifolds

We show that the geodesic flow of a compact Finsler manifold without conjugate points is transitive provided that the universal covering satisfies the uniform Finsler visibility condition. This result is a nontrivial extension of a well known theorem due to Eberlein for Riemannian manifolds. For doing so, we introduce suitable Finsler versions of the concepts of Gromov's δ-hyperbolicity and Eberlein's visibility, and study their consequences.     

On the measures of maximal entropy for the geodesic flow

Let M be a closed and connected manifold equipped with a C^∞ Riemannian metric. Using the geodesic arc formalism developed by Maé (J. Differential Geom. 45 (1997) 74–93) and Paternain [Ann. Sci. École Norm. Sup. 4eme série t 33 (2000) 121–138], we give a way of constructing a measure of maximal entropy for the geodesic flow.     

Ergodic measures of geodesic flows on compact Lie groups

Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature (possibly strictly positive on some sections), whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.     

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

Topological sequence entropy for maps of the interval

A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

     

Topological entropy and blocking cost for geodesics in Riemannian manifolds

For a pair of points x, y in a compact, Riemannian manifold M let n _t (x, y) (resp. s _t (x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block these geodesic segments). We study relationships between the growth rates of n _t (x, y) and s _t (x, y) as t → ∞. We obtain lower bounds on s _t (x, y) in terms of the topological entropy h(M) and the fundamental group π ₁(M). For instance, we show that if h(M) > 0 then s _t grows exponentially, with the rate at least h(M)/2. This strengthens earlier results on blocking of geodesics (Burns and Gutkin Discrete Contin Dyn Syst 21:403–413, 2008; Lafont and Schmidt Geom Topol 11:867–887, 2007), and puts them in a new perspective.      

拓扑熵的一个下界估计

设Ｘ为局部闭路可缩的紧致空间，ｆ为Ｘ的自映射，ｈ（ｆ）为ｆ的拓扑熵，Ｒ∞（ｆ）为ｆ的渐近Ｒｅｉｄｅｍｅｉｓｔｅｒ数，则有ｈ（ｆ）≥ｌｏｇＲ∞（ｆ）．     

On the measure with maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials

The Teichmüller flow g _t on the moduli space of Abelian differentials with zeros of given orders on a Riemann surface of a given genus is considered. This flow is known to preserve a finite absolutely continuous measure and is ergodic on every connected component ? of the moduli space. The main result of the paper is that µ/µ(?) is the unique measure with maximal entropy for the restriction of g _t to ?. The proof is based on the symbolic representation of g _t .     

A topological classification of integrable geodesic flows on the two-dimensional sphere with an additional integral quadratic in the momenta

Summary The analytic expression for a Riemannian metric on a 2-sphere, having integrable geodesic flow with an additional integral quadratic in momenta, is given in [Ko1]. We give the topological classification, up to topological equivalence of Liouville foliations, of all such metrics. The classification is computable, and the formula for calculating the complexity of the flow is straightforward. We prove Fomenko's conjecture that, from the point of view of complexity, the integrable geodesic flows with an additional integral linear or quadratic in momenta exhaust “almost all” integrable geodesic flows on the 2-dimensional sphere.     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

关于代换系统拓扑序列熵的注记

Goodman证明了对两符号等长代换系统, 如果代换规则中0和1对应的词只有一个位置不同,那么对应的代换系统为null的, 即此系统沿着任意正整数序列的序列熵均为0. 在本文中, 我们针对系统的结构特征, 通过考察因子系统, 给出了此经典结果的另外一种证明. 同时, 对此类代换系统沿着给定序列的复杂性, 我们得到了比Goodman更为精确的估计.     

Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds

In the first part of this paper, we prove the sharp global Li‐Yau type gradient estimates for positive solutions to doubly nonlinear diffusion equation(DNDE) on complete Riemannian manifolds with nonnegative Ricci curvature. As an application, one can obtain a parabolic Harnack inequality. In the second part, we obtain a Perelman‐type entropy monotonicity formula for DNDE on compact Riemannian manifolds with nonnegative Ricci curvature. These results generalize some works of Ni (JGA 2004), Lu–Ni–Vázquez–Villani (JMPA 2009) and Kotschwar–Ni (Annales Scientifiques de l'École Normale Supérieure 2009). Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic Formulae for Brillouin Index on Riemannian Manifolds

We establish several asymptotic formulae for Brillouin index on fiat tori. As an application of these formulae it is proved that the topological entropy of a geodesic flow on a fiat torus is zero.     

Hardy-type spaces for eigenfunctions of invariant differential operators on homogeneous line bundles over Hermitian symmetric spaces

We prove a Fatou-type theorem on a homogeneous line bundle over a Hermitian symmetric space, and characterize the range of the Poisson transform of L_p -functions on the maximal boundary as a Hardy-type space.     

Topological classification of supertransitive flows on closed nonorientable surfaces. Part I. Necessary conditions for topological equivalence

In this paper a topological invariant is introduced for the class of supertransitive flows on closed nonorientable surfacesM of negative Euler characteristic. We describe properties of this invariant and prove that it provides necessary conditions for the topological equivalence of flows belonging to the above-mentioned class of supertransitive flows. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 461–470, September, 2000.