Ratner's theorem on SL(2,R)-invariant measures

Joining measures for horocycle flows on abelian covers

A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. Recently, Mohammadi and Oh extended Ratner's theorem to horocycle flows on hyperbolic surfaces of infinite area but finite genus. In this paper, we present the first joining classification result of a horocycle flow on a hyperbolic surface of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface. We also discuss several applications.

Locally unipotent invariant measures and limit distribution of a sequence of polynomial trajectories on homogeneous spaces

Let $G$ be a Lie group and $Γ$ be a lattice in $G$. We introduce the notion of locally unipotent invariant measures on $G/Γ$. We then prove that under some conditions, the limit measure supported on the image of polynomial trajectories on $G/Γ$ is locally unipotent invariant, thus give a partial answer to an equidistribution problem for higher dimensional polynomial trajectories on homogeneous spaces, which was raised by Shah in \cite{shah1994limit}. The proof relies on Ratner's measure classification theorem, linearization technique for polynomial trajectories near singular sets and a twistin…

Invariant and stationary measures for the SL(2,R) action on Moduli space

We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner's seminal work.

Billiards in rectangles with barriers

We use Ratner's theorem to compute the asymptotics of the number of (cylinders of) periodic trajectories in a rectangle with a barrier, assuming that the location p/q of the barrier is rational. We also show that as q tends to infinity, the constant in the asymptotic formula tends to the constant for the generic genus 2 flat surface.

Characters of algebraic groups over number fields

Let $k$ be a number field, $\mathbf{G}$ an algebraic group defined over $k$, and $\mathbf{G}(k)$ the group of $k$-rational points in $\mathbf{G}.$ We determine the set of functions on $\mathbf{G}(k)$ which are of positive type and conjugation invariant, under the assumption that $\mathbf{G}(k)$ is generated by its unipotent elements. An essential step in the proof is the classification of the $\mathbf{G}(k)$-invariant ergodic probability measures on an adelic solenoid naturally associated to $\mathbf{G}(k);$ this last result is deduced from Ratner's measure rigidity theorem for homogeneous spa…

高秩中水平球面不变测度的分类：完整故事

Classification of horospherical invariant measures in higher rank: The Full Story

本文分类了任意半单实代数群的Anosov子群的水平球面不变Radon测度，将Burger和Roblin在秩一中的工作推广到高秩，并解决了Landesberg-Lee-Lindenstrauss-Oh和Oh提出的开放问题。

An ergodic theorem for non-invariant measures

Given a space $X$, a $σ$-algebra $\mathfrak{B}$ on $X$ and a measurable map $T:X \to X$, we say that a measure $μ$ is half-invariant if, for any $B \in \mathfrak{B}$, we have $μ(T^{-1}(B)\leq μ(B)$. In this note we present a generalization of Birkhoff's Ergodic theorem to $σ$-finite half-invariant measures.

A KAM-Theorem for Persistence of Quasi-periodic Invariant Tori in Bifurcation Theory of Equilibrium Points

In this paper, we establish a KAM-theorem for ordinary differential equations with finitely differentiable vector fields and multiple degeneracies. The theorem can be used to deal with the persistence of quasi-periodic invariant tori in multiple Hopf and zero-multiple Hopf bifurcations, as well as their subordinate bifurcations, of equilibrium points of continuous dynamical systems.

A dynamical version of Kuratowski-Mycielski Theorem and invariant chaotic sets

We establish a dynamical version of Kuratowski-Mycielski Theorem on the existence of "large" invariant dependent sets. We apply this result to the study of invariant chaotic sets in topological dynamical systems, simplify many known results on this topic and also obtain some new results.

A simple proof of the invariant torus theorem

We give a simple proof of Kolmogorov's theorem on the persistence of a quasiperiodic invariant torus in Hamiltonian systems. The theorem is first reduced to a well-posed inversion problem (Herman's normal form) by switching the frequency obstruction from one side of the conjugacy to another. Then the proof consists in applying a simple, well suited, inverse function theorem in the analytic category, which itself relies on the Newton algorithm and on interpolation inequalities. A comparison with other proofs is included in appendix.

Random invariant densities for Markov operator cocycles and random mean ergodic theorem

In the present paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.

Comparison theorems for invariant measures of random dynamical systems

We study a random dynamical system such that one transformation is randomly selected from a family of transformations and then applied on each iteration. For such random dynamical systems, we consider estimates of absolutely continuous invariant measures. Since the random dynamical systems are made by complicated compositions of many deterministic maps and probability density functions, it is difficult to estimate the invariant measures. To get rid of this difficulty, we present fundamental comparison theorems which make easier the estimates of invariant measures of random maps. We also demons…

General KAM theorems and their applications to invariant tori with prescribed frequencies

In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some non-degeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.

Prevalent behavior and almost sure Poincare-Bendixson Theorem for smooth flows with invariant k-cones

We investigate the global dynamics from a measure-theoretic perspective for smooth flows with invariant cones of rank k. For such systems, it is shown that prevalent (or equivalently, almost all) orbits will be pseudo-ordered or convergent to equilibria. This reduces to Hirsch's prevalent convergence Theorem if the rank k=1; and implies an almost-sure Poincare-Bendixson Theorem for the case k=2. These results are then applied to obtain an almost sure Poincare-Bendixson theorem for high-dimensional differential equations.

Kolmogorov's Theorem for Low-Dimensional Invariant Tori of Hamiltonian Systems

In this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the function $Ψ$ of two variables defined on semi-cylinder is established. It is proved that if unperturbed Hamiltonian has a saddle point, then under arbitrary perturbations there persists at least one hyperbolic torus.

科尔莫戈罗夫不变 torus 定理用于弱相互作用粒子 I：全维 tori

Kolmogorov invariant torus theorem for weakly interacting particles I: Full dimensional tori

本文提出一个抽象KAM定理，用于研究无限多相互作用粒子系统，构建全维KAM tori以应对长程相互作用。

Attractors with Non-Invariant Interior and Pinheiro's Theorem A

This is a provisional version of an article, intended to be devoted to properties of attractor's intertior for smooth maps (not diffeomorphisms). We were originally motivated for this research by Pinhero's Theorem A from his recent preprint, and in Section 3 we give a simple and straightforward proof of this result.

A complementary proof of Baker's theorem of completely invariant components for transcendental entire functions

Baker proved that for transcendental entire functions there is at most one completely invariant component of the Fatou set. It was observed by Julien Duval that there is a missing case in Baker's proof. In this article we follow Baker's ideas and give some alternative arguments to establish the result.

Existence of random invariant periodic curves via random semiuniform ergodic theorem

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

A proof of the invariant torus theorem of Kolmogorov

Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.

The Liouville-Arnold-Nekhoroshev theorem for non-compact invariant manifolds

Under ceratin conditions, generalized action-angle coordinates can be introduced near non-compact invariant manifolds of completely and partially integrable Hamiltonian systems.

On persistence of invariant tori and a theorem by Nekhoroshev

We give a proof of a theorem by N.N. Nekhoroshev concerning Hamiltonian systems with $n$ degrees of freedom and $s$ integrals of motion in involution, where $1 \le s \le n$. Such a theorem ensures persistence of $s$-dimensional invariant tori under suitable nondegeneracy conditions generalizing Poincaré's condition on the Floquet multipliers.

Linearized stability theorem for invariant and quasi-invariant parabolic differential equations in Banach manifolds with applications to free boundary problems

If a differential equation in a Banach manifold is invariant or quasi-invariant under the action of one or more Lie groups, then its stationary points cannot be isolated, so that classical linearized stability theorem does not apply to it. The first main purpose of this paper is to establish a linearized stability theorem for parabolic differential equations in Banach manifolds which are either invariant or quasi-invariant under actions of a number of Lie groups. The second purpose of this paper is to apply this theorem to analyze stability of stationary solutions of some free boundary problem…