Unconditional Proof of the Boltzmann-Sinai Ergodic Hypothesis

Certain systems of three falling balls satisfy the Chernov-Sinai ansatz

The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. Since almost three decades, the question of ergodicity is still open. The subject of this work is to contribute to the solution of the ergodicity conjecture for three falling balls with a specific mass ratio. The latter is executed in the following three points: First, we prove the Chernov-Sinai ansatz. Second, we prove that there is an abundance of least expanding points and, third, we explain that the proper alignment condition can still be verifi…

The Boltzmann-Sinai Ergodic Hypothesis in Two Dimensions (Without Exceptional Models)

We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^ν$, $ν\ge2$. In the case $ν=2$ we prove (the full hyperbolicity and) the ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, without exceptional values. In higher dimensions, for hard ball systems in $\Bbb T^ν$ ($ν\ge3$), we prove that every such system (is fully hyperbolic and) has open ergodic components.

遍历平均与勒贝格导数级数的绝对收敛与无条件收敛

Absolute and Unconditional Convergence of Series of Ergodic Averages and Lebesgue Derivatives

该研究探讨遍历理论与调和分析中各类随机过程级数的绝对及无条件收敛条件，涉及Cesàro平均、Lebesgue导数等对象的平均差相关级数。

Certain systems of three falling balls satisfy the Chernov-Sinai Ansatz

The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. For almost three decades new, the question of its ergodicity remains open. We contribute to the solution of the erogodicity conjecture for three falling balls with a specific mass ratio in the following two points: First, we prove the Chernov-Sinai ansatz. Second, we prove that there is an abundance of sufficiently expanding points. It is of special interest that for the aforementioned specific mass ratio, the configuration space can be unfolded to …

Flow-invariant hypersurfaces in semi-dispersing billiards

This work results from our attempts to solve Boltzmann-Sinai's hypothesis about the ergodicity of hard ball gases. A crucial element in the studies of the dynamics of hard balls is the analysis of special hypersurfaces in the phase space consisting of degenerate trajectories (which lack complete hyperbolicity). We prove that if a flow-invariant hypersurface $J$ in the phase space of a semi-dispersing billiard has a negative Lyapunov function, then the volume of the forward image of $J$ grows at least linearly in time. Our proof is independent of the solution of the Boltzmann-Sinai hypothesis, …

Upgrading the Local Ergodic Theorem for planar semi-dispersing billiards

The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional bill…

Singularities and nonhyperbolic manifolds do not coincide

We consider the billiard flow of elastically colliding hard balls on the flat $ν$-torus ($ν\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann-Sinai Ergodic Hypothesis.

Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions

The ultimate goal of our book is to present a unified approach to the dynamics, ergodic theory, and geometry of elliptic functions from $\C$ to $\oc$. We consider elliptic functions as a most regular class of transcendental meromorphic functions. Poles form an essential feature of such functions but the set of critical values is finite and an elliptic function is "the same" on its of its fundamental regions. In a sense this is the class of transcendental meromorphic functions which resembles rational functions most. On the other hand, the differences are huge. We will touch on them in the cour…

Wojtkowski落体粒子猜想的证明

Proof of Wojtkowski's Falling Particle Conjecture

本文通过新的代数方法，无条件证明了Wojtkowski关于一维弹性球落体系统遍历性的猜想，即几乎所有此类系统都是完全双曲且遍历的。

Extremes and extremal indices for level set observables on hyperbolic systems

Consider an ergodic measure preserving dynamical system $(T,X,μ)$, and an observable $ϕ:X\to\mathbb{R}$. For the time series $X_n(x)=ϕ(T^{n}(x))$, we establish limit laws for the maximum process $M_n=\max_{k\leq n}X_k$ in the case where $ϕ$ is an observable maximized on a curve or submanifold, and $(T,X,μ)$ is a hyperbolic dynamical system. Such observables arise naturally in weather and climate applications. We consider the extreme value laws and extremal indices for these observables on Anosov diffeomorphisms, Sinai dispersing billiards and coupled expanding maps. In particular we obtain clu…