Smooth mixing flows with singular spectra

Ratner's property and mild mixing for smooth flows on surfaces

Let $T=(T_t^f)_{t\in \mathbb{R}}$ be a special flow built over an IET $T : T \to T$ of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T. We show that $T$ satisfies so-called switchable Ratner's property. A consequence of this fact is that such flows are mildly mixing. Thus, on each compact, connected, orientable surface of genus greater than one there exist flows which are mildly mixing and not mixing.

奇异斜积的指数混合

Exponential mixing for singular skew-products

本文研究了非一致扩张映射与分段光滑映射构成的奇异斜积系统，证明了在允许纤维映射奇异行为的温和假设下，该映射相对于唯一的 SRB 测度呈指数混合，扩展了已有结论。

Exponential mixing, KAM and smooth local rigidity

Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.

Exponential Mixing and Smooth Classification of Commuting Expanding Maps

We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We…

Bifurcations of Non Smooth Vector Fields on $\R^2$ by Geometric Singular Perturbations

Our object of study is non smooth vector fields on $\R^2$. We apply the techniques of geometric singular perturbations in non smooth vector fields after regularization and a blow$-$up. In this way we are able to bring out some results that bridge the space between non$-$smooth dynamical systems presenting typical singularities and singularly perturbed smooth systems.

Mixing for Smooth Time-Changes of General Nilflows

We consider completely irrational nilflows on any nilmanifold of step at least $2$. We show that there exists a dense set of smooth time-changes such that any time-change in this class which is not measurably trivial gives rise to a mixing nilflow. This in particular reproves and generalizes to any nilflow (of step at least $2$) the main result proved in [AFU] for the special class of Heisenberg (step $2$) nilflows, and later generalized in [Rav2] to a class of nilflows of arbitrary step which are isomorphic to suspensions of higher-dimensional linear toral skew-shifts.

Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces

Given a compact surface $\mathcal{M}$ with a smooth area form $ω$, we consider an open and dense subset of the set of smooth closed 1-forms on $\mathcal{M}$ with isolated zeros which admit at least one saddle loop homologous to zero and we prove that almost every element in the former induces a mixing flow on each minimal component. Moreover, we provide an estimate of the speed of the decay of correlations for smooth functions with compact support on the complement of the set of singularities. This result is achieved by proving a quantitative version for the case of finitely many singularities…

Flexibility of statistical properties for smooth systems satisfying the central limit theorem

In this paper we exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties: (1) zero entropy; (2) weak but not strong mixing; (3) (polynomially) mixing but not $K$; (4) $K$ but not Bernoulli; (5) non Bernoulli and mixing at arbitrary fast polynomial rate. We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index $1$.

Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

We consider typical area preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affimatively to Rohlin's multiple mixing question in this context. The main tool is a variation of the Ratner property (a property originally proved by Ratner for the horocycle flow), i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behaviour of these flows and has impli…

Strongly mixing smooth planar vector field without asymptotic directions

We use a Voronoi-type tessellation based on a compound Poisson point process to construct a polynomially mixing stationary random smooth planar vector field with bounded nonnegative components such that, with probability one, none of the associated integral curves possess an asymptotic direction.

时间变换的水平流混合理论

Mixing asymptotics for time-changes of horocycle flows

本文研究了光滑时间变换的水平流的混合性质，通过改进的剪切混合方法，证明了多项式上界与Ratner的最优速率一致，并在存在谱间隙时建立了精确的多项式渐近性。

射影化导数扩张的弱混合行为

Weak mixing behavior for the projectivized derivative extension

本文在光滑和解析范畴中构造了拓扑熵为零、其微分在切丛射影化中关于光滑测度弱混合的二维流形微分同胚，核心方法是定量版共轭逼近技术。

抛物系统的多重混合

Multiple mixing for parabolic systems

本文引入局部一致剪切系统（LUS）以严格描述剪切混合机制，证明所有 LUS 流具有任意阶混合性质，并证实曲面光滑流及 unipotent flow 的时间变换满足多项式任意阶混合。

类 Lelek 扇：支撑拓扑混合映射的端点稠密连续统

Lelek-like Fans: Endpoint-dense Continua Supporting Topologically Mixing Maps

本文构造了一族两两互不同胚且端点稠密的非光滑扇，并证明其中每个扇都支撑拓扑混合的非可逆映射和拓扑混合同胚。

Weakly mixing smooth planar vector field without asymptotic directions

We construct a planar smooth weakly mixing stationary random vector field with nonnegative components such that, with probability 1, the flow generated by this vector field does not have an asymptotic direction. Moreover, for all individual trajectories, the set of partial limiting directions coincides with those spanning the positive quadrant. A modified example shows that a particle in space-time weakly mixing positive velocity field does not necessarily have an asymptotic average velocity.

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

Denseness of robust exponential mixing for singular-hyperbolic attracting sets

There exists a $C^2$-open and $C^1$-dense subset of vector fields exhibiting singular-hyperbolic attracting sets (with codimension-two stable bundle), in any $d$-dimensional compact manifold ($d\ge3$), which mix exponentiallu with respect to any physical/SRB invariant probability measure. More precisely, we show that given any connected singular-hyperbolic attracting set for a $C^2$-vector field $X$, there exists a $C^1$-close multiple of $X$ of class $C^2$, generating a topologically equivalent flow, which is robustly exponentially mixing with respect to any physical measure for all vector fi…

Weakly Mixing Systems with Dense Prime Orbits

We show existence of smooth, weakly mixing reparametrizations of some linear flows on $\mathbb{T}^2$ for which all orbits sampled at prime times are dense.

Weak mixing for interval exchange transformations and translation flows

We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$ (with prescribed singularity types) is weakly mixing.

Weakly mixing diffeomorphisms preserving a measurable Riemannian metric are dense in $\mathcal{A}_α\left(M\right)$ for arbitrary Liouvillean number $α$

We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing $C^{\infty}$-diffeomorphisms which preserve both a smooth volume $ν$ and a measurable Riemannian metric is dense in $\mathcal{A}_α \left(M\right)= \overline{ \left\{h \circ S_α \circ h^{-1} : h \in \text{Diff}^{\infty}\left(M, ν\right) \right\}}^{C^{\infty}}$ for every Liouvillean number $α$. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly c…

Robust Exponential Mixing and Convergence to Equilibrium for Singular Hyperbolic Attracting Sets

We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and finitely many ergodic physical/SRB invariant probability measures, whose basins cover a full Lebesgue measure subset of the trapping region of the attracting set. We obtain exponential mixing for any physical probability measure supported in the trapping region and also exponential convergence to equilibrium, for a $C^2$ open subset of vector fields in any $d$-…

Integrability and adapted complex structures to smooth vector fields on the plane

Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields $\mathbb{X}$ and smooth vector fields $X$. Our approximation route studies three integrability notions for real smooth vector fields $X$ with singularities on the plane or the sphere. The first notion is related to Cauchy-Riemann equations, we say that a vector field $X$ admits an adapted complex structure $J$ if there exists a singular complex analytic vec…

Mixing for Time-Changes of Heisenberg Nilflows

We consider reparametrizations of Heisenberg nilflows. We show that if a Heisenberg nilflow is uniquely ergodic, all non-trivial time-changes within a dense subspace of smooth time-changes are mixing. Equivalently, in the language of special flows, we consider special flows over linear skew-shift extensions of irrational rotations of the circle. Without assuming any Diophantine condition on the frequency, we define a dense class of smooth roof functions for which the corresponding special flows are mixing. Mixing is produced by a mechanism known as stretching of Birkhoff sums. The complement o…

On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in $\mathbb{R}^3$

In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of singularities of piecewise smooth dynamical systems \cite{filippov1988differential} in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a simple manner. The…

Continuous spectrum for a class of smooth mixing Schrödinger operators

We give the first example of a smooth volume preserving mixing dynamical system such that the discrete Schrödinger operators on the line defined with a potential generated by this system and a Hölder sampling function, have almost surely a continuous spectrum.

Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces

We establish exponential mixing for the geodesic flow $φ_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the pha…

Exponential mixing for random dynamical systems and an example of Pierrehumbert

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mi…

任意空间维度下光滑速度场的最优混合与增强耗散

Optimal mixing and enhanced dissipation for a smooth velocity field in any spatial dimension

本文在任意维度环面上构造光滑随机速度场，证明其作为通用指数混合器，并在小分子扩散下以最优速率增强耗散，方法基于随机动力系统和两点马尔可夫过程的均匀遍历性。

Singularity of the spectrum for smooth area-preserving flows in genus two and translation surfaces well approximated by cylinders

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non degenerate isomorphic saddle has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for…