Nurlan S. Dairbekov, Gabriel P. Paternain
We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $θ$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ π+θ\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $θ$ is closed. This result was proved in \cite{DP2} under the assumption that the flow of $\G$ is Anosov.
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In this paper we give sufficient conditions for existence of a solution of cohomological equation for suspension flows over automorphisms of Markov compacta, which were introduced by Vershik and Ito. The main result (Theorem 1) can be regarded as a symbolic analogue of results due to Forni and Marmi, Moussa and Yoccoz for translation flows and interval exchange transformations.
Cohomological equation for geodesic flows on flat surfaces
该研究针对带有限锥点的紧致平坦曲面，证明了其单位切丛上测地流的上同调方程解的存在性，还得到非有理完整乐群曲面的完整乐叶状结构遍历性及水平叶状拉普拉斯算子的无上同调性质。
We study the twisted cohomoligical equation over the geodesic flow on $SL(2,\mathbb{R})/Γ$. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sobolev norms) between the twisted coboundary and the solution. We also give a tame splittings for non-homogeneous cohomological equations. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for partially hyperbolic actions in …
Twisted cohomological equations for translation flows
通过调和分析证明扭曲上同调方程解的存在性，导数损失至多3+，进而证明乘积平移流及时间-τ映射的稳定性。
Let $\mathbb{G}$ be a higher-rank connected semisimple Lie group with finite center and without compact factors. In any unitary representation $(π, \mathcal{H})$ of $\mathbb{G}$ without non-trivial $\mathbb{G}$-fixed vectors, we study the twisted cohomological equation $(X+m)f=g$, where $m\in\mathbb{R}$ and $X$ is in a $\mathbb{R}$-split Cartan subalgebra of $\text{Lie}(\mathbb{G})$. We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for $f$. We also study common solution to (the inf…
In this paper we introduce a new compactness condition - Property (C) - for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient f…
These notes are based on lectures given by the author at the Summer School on Teichmüller dynamics, mapping class groups and applications in Grenoble, France, in June 2018 and at the Oberwolfach Seminar on Anisotropic Spaces and their Applications to Hyperbolic and Parabolic Systems in June 2019. We derive results about the so-called Ruelle resonances and the asymptotics of correlations for several classes of systems from known results on cohomological equations and invariant distributions for the respective unstable vector fields. In particular, we consider pseudo-Anosov diffeomorphisms on su…
Solving the cohomological equation for locally hamiltonian flows, part I -- local obstructions
本文研究紧致曲面上局部哈密顿流的余同调方程，通过定义新的不变分布族（局部障碍）并分析其影响，给出了解的正则性条件，为完整解决该问题提供关键要素。
We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of $\mathbb{R}^{d}$. We use this to characterize closed negatively curved Rie…
Let X be a vector field on a compact connected manifold M. An important question in dynamical systems is to know when a function g:M -> R is a coboundary for the flow generated by X, i.e. when there exists a function f: M->R such that Xf=g. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions D_n such that any sufficiently smooth function g is a coboundary iff it belongs to the kernel of all the distributions D_n.
Solving the cohomological equation for locally hamiltonian flows, part II -- global obstructions
本文研究紧曲面上局部哈密顿流的上同调方程，在完全滤过丢番图条件下构造全局不变分布，结合局部分布量化解的最优正则性，并得到 Kontsevich-Zorich 余循环的谱结果。
Effective Unique Ergodicity and Weak Mixing of Translation Flows
本文介绍基于Hodge理论的上同调方法研究平移流的有效唯一遍历性与弱混合，首次用扭上同调Hodge理论给出Veech弱混合准则的新证明，并提供相关遍历理论的入门内容与研究问题。
We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation surfaces our results are sharp and we find the expected relation between the regularity of the distributional obstructions and the Lyapunov exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we exactly determine the dimension of the space of obstructions in each Sobolev regularity class in terms of the Kontsevich-Zorich exponents. For…
We study the cohomological pressure introduced by R.Sharp (defined by using topological pressures of certain potentials of Anosov flows). In particular, we get the rigidity in the case that this pressure coincides with the metrical entropy, generalising related rigidity results of A.Katok and P. Foulon.
This article surveys recent progress of results in topology and dynamics based on techniques of closed one-forms. Our approach allows us to draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically we describe a Lusternik - Schnirelmann type theory for closed one-forms, the focusing effect for flows and the theory of Lyapunov one-forms. We also discuss recent results about cohomological treatment of the invariants cat(X, ξ) and cat^1(X, ξ) and their explicit computation in certain examples.
We consider parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeormorphisms. This class of flows includes nilflows on the Heisenberg nilmanifold which are renormalized by partially hyperbolic automorphisms. The transfer operators associated to the renormalization maps, acting on anisotropic Sobolev spaces, are known to have good spectral properties (this relies on ideas which have some resemblance to representation theory but also apply to non-algebraic systems). The spectral information is used to describe the deviation of ergodic averages and…
We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation $Ψ-Ψ\circ T=Φ$ has a bounded solution $Ψ$ provided that the datum $Φ$ belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The class of interval exchange maps is characterized in terms of a diophantine condition of ``Roth type'' imposed to an acceleration of the Rauzy--Veech--Zorich continued fraction expansion associated to T. CONTENTS 0. Introduction 1. The continued fraction algorithm for interval exchange…
We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.
Tensor tomography and frame flow ergodicity for magnetic flows in higher dimensions
本文将测地流理论中的两个结果扩展到任意维流形上的磁流，研究磁射线变换并建立张量断层成像结果，分析磁框架流在紧缩条件下的遍历性，发展了适应磁流的黎曼几何概念。
Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $Ω$ be a 2-form. We show that the magnetic flow of the pair $(g,Ω)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $Ω$ is a constant multiple of the area form of $g$ and the magnetic flow is a horocycle flow. This characterization of horocycle flows implies that if the magnetic flow of a pair $(g,Ω)$ is $C^1$-conjugate to the horocycle flow of a hyperbolic metric $\bar{g}$ then there exists a constant $a>0$, such that $ag$ and $\bar{g}$ are isometric an…
Magnetic Killing tensors and first integrals of the magnetic flow
本文提出磁 Killing 对称张量，用于构造磁流的首个积分，并证明某些 nilmanifolds 上磁流的可积性。
Anosov magnetic flows on surfaces
本文利用 Wojtkowski 的商丛方法，给出了闭定向曲面上磁流成为 Anosov 流的充要条件。
We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. This is a magnetic analog of Hilbert's fourth problem asking to describe the Finsler metrics whose geodesics are straight lines.
Magnetic flows on 3D contact sub-Riemannian manifolds via the Rumin complex
本文通过Rumin复形研究三维接触亚黎曼流形上的磁流，揭示其与提升的亚黎曼结构之间的关系，并分析磁场对结构几何的影响。
We consider magnetic flows on 2-step nilmanifolds $M = Γ\backslash G$, where the Riemannian metric $g$ and the magnetic field $σ$ are left-invariant. Our first result is that when $σ$ represents a rational cohomology class and its restriction to $\mathfrak{g} = T_eG$ vanishes on the derived algebra, then the associated magnetic flow has zero topological entropy. In particular, this is the case when $σ$ represents a rational cohomology class and is exact. Our second result is the construction of a magnetic field on a 2-step nilmanifold that has positive topological entropy for arbitrarily high …
Let $M$ be a closed oriented surface of negative Gaussian curvature and let $Ω$ be a non-exact 2-form. Let $λ$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la Ω$ is a coboundary if and only if the Gaussian curvature is constant and $Ω$ is a constant multiple of the area form.
Final version. To appear in Discrete and Continuous Dynamical Systems - A.
Guillarmou's Normal Operator for Magnetic and Thermostat Flows
本文研究了磁流和热疗流的正常算子，证明其为椭圆伪微分算子，并给出了磁X射线变换的稳定性估计。
We study the topological entropy of the magnetic flow on a closed riemannian surface. We prove that if the magnetic flow has a non-hyperbolic closed orbit in some energy set T^cM= E^{-1}(c), then there exists an exact $ C^\infty$-perturbation of the 2-form $ Ω$ such that the new magnetic flow has positive topological entropy in T^cM. We also prove that if the magnetic flow has an infinite number of closed orbits in T^cM, then there exists an exact C^1-perturbation of $ Ω$ with positive topological entropy in T^cM. The proof of the last result is based on an analog of Franks' lemma for magnetic…
We obtain Kupka-Smale theorem and Franks lemma for magnetic flows on manifolds with any dimension. This improves Miranda result on surfaces. However our methods relies on geometric control theory, like in Rifford and Ruggiero articles.
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。
In this paper the magnetic geodesic flow on a 2-torus is considered. We study a semi-hamiltonian quasi-linear PDEs which is equivalent to the existence of polynomial in momenta first integral of magnetic geodesic flow on fixed energy level. It is known that diagonal metric associated with this system is Egorov one if degree of the first integral is equal to 2 or 3. In this paper we prove this fact in the case of existence of the first integral of any degree.
We completely integrate the magnetic geodesic flow on a flat two-torus with the magnetic field $F = \cos (x) dx \wedge dy$ and describe all contractible periodic magnetic geodesics. It is shown that there are no such geodesics for energy $E \geq 1/2$, for $E< 1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.
Integrability of the magnetic geodesic flow on the sphere with a constant 2-form
研究球面上磁测地流的可积性，证明其为Liouville可积，积分形式为二次和线性动量函数。
In this paper the geodesic flow on a 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral $F$ on $N+2$ different energy levels which is polynomial in momenta of arbitrary degree $N$ with analytic periodic coefficients. It is proved that in this case the magnetic field and metrics are functions of one variable and there exists a linear in momenta first integral on all energy levels.
In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
We describe a class of completely integrable $G$-invariant magnetic geodesic flows on (co)adjoint orbits of a compact connected Lie group $G$ with magnetic field given by the Kirillov-Konstant 2-form.
We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable.
It is proved that the motion of a charge particle on a hyperbolic oriented two-dimensional surface in a magnetic field given by the volume form of the hyperbolic metric is completely integrable on the energy levels E < 1/2 in terms of real-analytic integrals. However it was known that on the level E=1/2 every trajectory is transitive and the restrictions of this flow onto the levels E>1/2 are Anosov flows
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