Damien Thomine
We construct families of two-dimensional Sinai billiards whose transfer operators have Ruelle resonances arbitrarily close to 1. Our method involves taking a large enough cover of an initial billiard table, and relating the transfer operator of the covering table to twisted transfer operators of the initial table. We also study the distribution of these resonances which are close to 1.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Sinai-Ruelle-Bowen measures are the only physically observable invariant measures for billiard dynamical systems under small perturbations. These measures are singular, but as it was observed, marginal distributions of spatial and angular coordinates are absolutely continuous. We generalize these facts and provide full mathematical proofs.
In this article, we recover the Sinai - Ruelle - Bowen measure associated to a real-valued Hölder continuous function defined on the Julia set of a hyperbolic quadratic polynomial, as a noncommutative measure by constructing an appropriate spectral triple.
An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties, paying special attention to the geometric `push-forward' approach. After describing this approach in the uniformly hyperbolic setting, we review recent work that extends it to non-uniformly hyperbolic systems.
The Bernoulli property of Sinai-Ruelle-Bowen measures for flows
研究$C^{1+β}$流的弱混合超几何SRB测度的伯努利性质，证明其具有伯努利性质。
The space of convex projective structures has been well studied with respect to the topological entropy. But, to better understand the geometry of the structure, we study the entropy of the Sinai-Ruelle-Bowen measure and show that it is a continuous function.
We study the rate of mixing of observables of Z^d-extensions of probability preserving dynamical systems. We explain how this question is directly linked to the local limit theorem and establish a rate of mixing for general classes of observables of the Z^2-periodic Sinai billiard. We compare our approach with the induction method.
A class of piecewise $C^2$ Lozi-like maps in three-dimensional Euclidean spaces is introduced, and the existence of Sinai-Ruelle-Bowen measures is studied, where the dimension of the instability is equal to two. Further, an example with computer simulations is provided to illustrate the theoretical results.
Linear response for Sinai billiards with small holes
研究带边界小孔的 Sinai 台球系统，证明条件生存概率测度关于孔尺寸 t 在 t=0 处可微，并计算其导数。
We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers with exponential tails. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in $(1,2)$.
Every finite horizon Sinai billiard map has a unique measure of maximal entropy
本文研究了有限时间辛恩尼亚德映射的最大熵测度存在性和唯一性，通过新的证明方法不依赖稀疏复发条件，证明了其唯一性。
Sinai-Ruelle-Bowen measure Entropy of geodesic flow on Convex Projective Surfaces
该研究聚焦凸实射影曲面上测地流的Sinai-Ruelle-Bowen测度熵，推导了熵与Hilbert面积、Blaschke度量下面积的极限关联特性。
The Lorentz gas of $\mathbb{Z}^2$-periodic scatterers (or the so called Sinai billiards) can be used to model motion of electrons on an ionized medal. We investigate the linear response for the system under various external forces (during both the flight and the collision). We give some characterizations under which the forced system is time-reversible, and derive an estimate of the electrical current generated by the forced system. Moreover, applying Pesin entropy formula and Young dimension formula, we get several characterizations of the non-equilibrium steady state of the forced system.
Markov partitions designed by Sinai(1968) and Bowen(1970) proved to be an efficient tool for descibing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular, for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai(1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich-Chernov-Sinai, 1990), Markov towers (Young, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai's original idea is surveyed i…
A generalized push-forward construction of Sinai-Ruelle-Bowen measures
本文针对正体积集合上有效双曲的微分同胚，提出SRB测度构造中参考测度的广义选取方法，其结果还适用于均匀部分双曲或具主导分裂的微分同胚。
We investigate the question of the rate of mixing for observables of a Z d-extension of a probability preserving dynamical system with good spectral properties. We state general mixing results, including expansions of every order. The main part of this article is devoted to the study of mixing rate for smooth observables of the Z 2-periodic Sinai billiard, with different kinds of results depending on whether the horizon is finite or infinite. We establish a first order mixing result when the horizon is infinite. In the finite horizon case, we establish an asymptotic expansion of every order, e…
A CAT(0)-approach to the marked length spectral rigidity of Sinai billiards
该研究针对有限视界Sinai台球的谱刚性问题，引入增强标记长度谱EL，结合CAT(0)空间方法与适配的Otal方法，证明同EL的Sinai台球等距，拓展了标记长度谱刚性理论的适用范围。
A direct proof of the existence of MME for finite horizon Sinai billiards
本文通过引入可忽略奇点条件，直接证明有限视界 Sinai 台球的度量压力上半连续性及 MME 存在性，并建立周期轨道等分布结果。
A family of natural equilibrium measures for Sinai billiard flows
本文研究了二维环面上 Sinai 台球流的平衡态，通过定义拓扑压力并利用各向异性 Banach 空间上的转移算子，证明了在充分条件下平衡态的唯一性、Bernoulli 性与流适应性，并给出了相关台球桌示例。
Statistical equilibration of energies in a slow-fast system is a fundamental open problem in physics. In a recent paper, it was shown that the equilibration rate in a springy billiard can remain strictly positive in the limit of vanishing mass ratio (of the particle and billiard wall) when the frozen billiard has more than one ergodic components [Proc. Natl. Acad. Sci. USA 114, E10514 (2017)]. In this paper, using the model of a springy Sinai billiard, it is shown that this can happen even in the case where the frozen billiard has a single ergodic component, but when the time of ergodization i…
We prove the abundance of Sinai-Ruelle-Bowen measures for diffeomorphisms away from ones with a homoclinic tangency. This is motivated by conjectures of Palis on the existence of physical (Sinai-Ruelle-Bowen) measures for global dynamics. The main novelty in this paper is that we have to deeply study Gibbs $cu$-states in different levels. Note that we have to use random perturbations to give some upper bound of the level of Gibbs $cu$-states.
For Dynamical Systems, a strong bound on multiple correlations implies the Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is derived for dynamically Holder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension.
Measure of maximal entropy for finite horizon Sinai billiard flows
基于Carrand的平衡态工作与“跳跃”自举方法，本文构造并证明了二维有限视野Sinai台球流在拓扑熵严格大于s_0 log 2条件下的唯一极大熵测度及其Bernoulli性质。
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, …
We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group L_t of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions. (Revised after referees' comments.)
We prove that the time of the first collision between two particles in a Sinai billiard table converges weakly to an exponential distribution when time is rescaled by the inverse of the radius of the particles. This results provides a first step in studying the energy evolution of hard ball systems in the rare interaction limit.
Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly d…
For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat (1998) for geodesic flows on compact surfaces (for general potentials)and transitive Anosov flows on compact manifolds with C^1 jointly non-integrable horocycle foliations (for the Sinai-Bowen-Ruelle potential). Here we deal with general potentials. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, clo…
We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. We also obtain an asymptotic estimate for the tail probability of the first return time to the initial cell. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight …
Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about mixing properties for billiard tables with non-compact cusps. There is no consensual definition of mixing for systems with infinite invariant measure. In this paper we study geometric and ergodic properties of billiard tables with a non-compact cusp. The goal of this text is, using the definition of mixing p…
We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.
On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps
本文针对有限视界 Sinai 台球映射提出拓扑熵定义 $h_*$，证明其与 Bowen 定义等价，并在稀疏回归条件下构造唯一且 Bernoulli 的最大熵测度，同时给出周期点数量的下界。
Adapted Measures for Markov Interval Maps
为研究熵与适配性的关联，本文考察含单个奇点的马尔可夫区间映射，证明映射熵与奇点“强度”的关联条件决定最大熵测度是否适配该奇点，并指出 Hölder 条件下奇点回复性是非适配不变测度存在的必要条件。
Locally Maximizing orbits for multi-dimensional Twist maps and Birkhoff billiards
本文研究多维辛扭转映射（特别是Birkhoff台球映射）的局部极大化轨道，提出几何准则并证明该映射的两个生成函数具有相同的局部极大化轨道类。
Transfer Operators and SRB Measures for Axiom A Diffeomorphisms: Spectral Gap, Structural Stability, and the Gibbs Equivalence Theorem
本文研究Axiom A微分同胚的转移算子和SRB测度，通过马尔可夫分区编码将符号谱结果推广到光滑动力学，提出四个主要定理，包括结构稳定性、转移算子的拟紧致性、SRB测度的构造以及佩辛熵公式，最终通过四个主要定理和导入结果证明吉布斯等价定理。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。