The Rotation Class of a Flow
Alex Clark
推广了A. Weil，我们引入了一个拓扑不变量，用于紧致的，连通的，有限维的，交换的，拓扑群上的流。我们计算这个不变量的一些例子，并与其他流不变量比较。
Generalizing a construction of A. Weil, we introduce a topological invariant for flows on compact, connected, finite dimensional, abelian, topological groups. We calculate this invariant for some examples and compare the invariant with other flow invariants.
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The liftable centralizer for special flows over irrational rotations is studied. It is shown that there are such flows under piecewise constant roof functions which are rigid and whose liftable centralizer is trivial.
A Dichotomy between Discrete and Continuous Spectrum for a Class of Special Flows over Rotations
我们提供了充分条件，使其相关的特殊流在任何无理旋转是弱混合或$L^2$共轭的悬浮液流。
Kakutani equivalence for products of some special flows over rotations
本文研究旋转上特殊流乘积的Kakutani等价性，通过估计Kakutani不变量，构造了可数无穷多个两两非等价的流乘积族。
Eigenvalues of transformations arising from irrational rotations and step functions. (Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier)
给定$\M T$上的无理旋转$T$，我们给出了上同调方程$$\exp{(2iπϕ)}=\e{2iπt}f/f\rond T.$$存在可测解的阶跃函数$ϕ$和$t\in \M T$的充要条件，得到了由无理旋转和阶梯函数引起的几种变换的特征值和特征函数的特征：柱流、特殊流、诱导映射和非线性流。从那里，我们给出特殊的流和三个区间交换变换与不寻常的谱性质的建设。在这两种情况下，我们展示了无限秩的克罗内克因子的例子。我们还构造了三个区间的交换变换，这些变换与无理旋转或里程计是非平凡共轭的。类似地，在无理旋转上也存在特殊的流，这些流与$\M T^2$上的平移流或在实数上的平移流非平凡共轭。最后，我们证明了一个正则化属性，使我们能够给出类似的例子，特殊的流动与光滑的天花板功能，自然丢番图条件下的旋转。
We consider special flows over two-dimensional rotations by $(α,β)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)\,dx\,dy\neq 0\neq \int_{\T^2}f_y(x,y)\,dx\,dy.$ Such flows are shown to be always weakly mixing and never partially rigid. For an uncountable set of $(α,β)$ with both $α$ and $β$ of unbounded partial quotients the strong mixing property is proved to hold. It is also proved that while specifying to a subclass of roof functions and to ergodic rotations for which $α$ and $β$ are of bounded partial quotients the correspon…
On mild mixing of special flows over irrational rotations under piecewise smooth functions
证明了在具有有界偏导数的无理$α$旋转上和在具有非零跳跃和的分段绝对连续的$f$下的所有特殊流都是温和混合的.这样的流量也示出享受一个条件，模仿拉特纳条件引入\cite{Rat}。因此，我们在$\T^2$上构造了一个具有一个奇点的光滑向量场，使得相应的流$(ϕ_t)_{t\in\R}$保持一个光滑测度，它的遍历分量集由一族周期轨道和一个正测度分量组成，在这个分量上$(ϕ_t)_{t\in\R}$是温和混合的，并且与所有混合流谱不相交.
We consider special flows over the rotation by an irrational $α$ under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and the sum of jumps $\neq 0$. We show that all such flows are weakly mixing. Under the additional assumption that $α$ has bounded partial quotients, we study weak Ratner's property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is classical that the flows under consideration are not mixing, one more condition on the s…
We consider special flows over the rotation on the circle by an irrational $α$ under roof functions of bounded variation. The roof functions, in the Lebesgue decomposition, are assumed to have a continuous singular part coming from a quasi-similar Cantor set (including the Devil's staircase case). Moreover, a finite number of discontinuities is allowed. Assuming that $α$ has bounded partial quotients, we prove that all such flows are weakly mixing and enjoy weak Ratner's property. Moreover, we provide a sufficient condition for the roof function to obtain a stability of the cocycle Ratner's pr…
Rotation vectors, as defined for homeomorphisms of the torus that are isotopic to the identity, are generalized to such homeomorphisms of any complete Riemannian manifold with non-positive sectional curvature. These generalized rotation vectors are shown to exist for almost every orbit of such a dynamical system with respect to any invariant measure with compact support. The concept is then extended to flows and, as an application, it is shown how non-null rotation vectors can be used to construct a measurable semi-conjugacy between a given flow and the geodesic flow of a manifold.
We prove that an analytic quasiperiodically forced circle flow with a not super-Liouvillean base frequency and which is close enough to some constant rotation is $C^{\infty}$ rotations reducible, provided its fibered rotation number is Diophantine with respect to the base frequency. As a corollary, we obtain that among such systems, the linearizable ones and those displaying mode-locking are locally dense for the $C^\infty$-topology.
By studying the weak closure of multidimensional off-diagonal self-joinings we provide a criterion for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid automorphisms. In particular, we apply the criterion to special flows over irrational rotations, providing a large class of non-reversible flows, including some analytic reparametrizations of linear flows on the two torus, so called von Neumann's flows and some special flows with piecewise polynomial roof functions.. A topological counterpart is also developed with th…
Local conjugacy classes for analytic torus flows
如果多维环面上的实解析流足够接近线性，有唯一的旋转向量满足算术条件Y，则它是解析共轭线性的。我们证明了这一点，通过证明下重整化的恒定Y向量场的轨道吸引所有附近的轨道具有相同的旋转矢量。
In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e., homotopically non-trivial invariant curves of the cylinder on which the dynamics is conjugate to a rotation) for any rational rotation number in an open interval (without any further assumption on its size), then the deformation must be trivial. Analogue phenomena for integrable Tonelli Hamiltonian flows in higher dimensions are discussed.
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