Ruelle Operator Theorem for Nonexpansive systems

Fixed points of the Ruelle-Thurston operator and the Cauchy transform

We give necessary and sufficient conditions for a function in a naturally appearing functional space to be a fixed point of the Ruelle-Thurston operator associated to a rational function, see Lemma 2.1. The proof uses essentially a recent [13]. As an immediate consequence, we revisit Theorem 1 and Lemma 5.2 of [11], see Theorem 1 and Lemma 2.2 below.

On Gibbs measures and spectra of Ruelle transfer operators

We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates.

关于Ruelle定理向长程势的一个推广

On an extension of a theorem by Ruelle to long-range potentials

本研究通过比较半直线与全直线上的Gibbs测度，提出寻找Ruelle转移算子本征函数的新方法，证明了含长程Dyson势在内的宽类势的可积本征函数存在性，并对其子类证明了本征函数的连续性。

Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case

Transfer operators M_k acting on k-forms in R^n are associated to smooth transversal local diffeomorphisms and compactly supported weight functions. A formal trace is defined by summing the product of the weight and the Lefschetz sign over all fixed points of all the diffeos. This yields a formal Ruelle-Lefschetz determinant Det^#(1-zM). We use the Milnor-Ruelle-Kitaev equality (recently proved by Baillif), which expressed Det^#(1-zM) as an alternated product of determinants of kneading operators,Det(1+D_k(z)), to relate zeroes and poles of the Ruelle-Lefschetz determinant to the spectra of th…

Spectral estimates for Ruelle operators with two parameters and sharp large deviations

We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + ıb)τ+ (c + ıd) g}$ with two complex parameters and Hölder functions $f,\: g$ generalizing the case $\Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].

A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces

We establish a maximin characterisation of the linear escape rate of the orbits of a non-expansive mapping on a complete (hemi-)metric space, under a mild form of Busemann's non-positive curvature condition (we require a distinguished family of geodesics with a common origin to satisfy a convexity inequality). This characterisation, which involves horofunctions, generalises the Collatz-Wielandt characterisation of the spectral radius of a non-negative matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981), concerning non-expansive maps in Banach spaces, a variant of a Denjoy-…

Spectral Properties of the Ruelle Operator on the Walters Class over Compact Spaces

Recently the Ruelle-Perron-Fröbenius theorem was proved for Hölder potentials defined on the symbolic space $Ω=M^{\mathbb{N}}$, where (the alphabet) $M$ is any compact metric space. In this paper, we extend this theorem to the Walters space $W(Ω)$, in similar general alphabets. We also describe in detail an abstract procedure to obtain the Fréchet-analyticity of the Ruelle operator under quite general conditions and we apply this result to prove the analytic dependence of this operator on both Walters and Hölder spaces. The analyticity of the pressure functional on Hölder spaces is established…

Around the nonlinear Ryll-Nardzewski theorem

Suppose that $Q$ is a weak$^{\ast }$ compact convex subset of a dual Banach space with the Radon-Nikodým property. We show that if $(S,Q)$ is a nonexpansive and norm-distal dynamical system, then there is a fixed point of $S$ in $Q$ and the set of fixed points is a nonexpansive retract of $Q.$ As a consequence we obtain a nonlinear extension of the Bader-Gelander-Monod theorem concerning isometries in $L$-embedded Banach spaces. A similar statement is proved for weakly compact convex subsets of a locally convex space, thus giving the nonlinear counterpart of the Ryll-Nardzewski theorem.

Duality between Eigenfunctions and Eigendistributions of Ruelle and Koopman operators via an integral kernel

We consider the classical dynamics given by a one sided shift on the Bernoulli space of $d$ symbols. We study, on the space of Hölder functions, the eigendistributions of the Ruelle operator with a given potential. Our main theorem shows that for any isolated eigenvalue, the eigendistributions of such Ruelle operator are dual to eigenvectors of a Ruelle operator with a conjugate potential. We also show that the eigenfunctions and eigendistributions of the Koopman operator satisfy a similar relationship. To show such results we employ an integral kernel technique, where the kernel used is the i…

扩张 Thurston 映射的素轨道定理：Lattès 映射与分裂 Ruelle 算子

Prime orbit theorems for expanding Thurston maps: Lattès maps and split Ruelle operators

本文为二维球面上的扩张 Thurston 映射建立了素数定理的类比，证明在满足 α-强不可积性条件的权重下，原始周期轨道数渐近服从对数积分并带指数误差界，结果覆盖后临界有限有理映射，并对 Lattès 映射给出更强结论。

Gibbs and equilibrium measures for some families of subshifts

For SFTs, any equilibrium measure is Gibbs, as long a $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobru{š}in states that for strongly-irreducible subshifts, shift-invariant Gibbs-measures are equilibrium measures. Here we prove a generalization of the Lanford-Ruelle theorem: for all subshifts, any equilibrium measure for a function with $d$-summable variation is "topologically Gibbs". This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium me…

Correlation Inequalities and Monotonicity Properties of the Ruelle Operator

Let $X = \{1,-1\}^\mathbb{N}$ be the symbolic space endowed with the product order. A Borel probability measure $μ$ over $X$ is said to satisfy the FKG inequality if for any pair of continuous increasing functions $f$ and $g$ we have $μ(fg)-μ(f)μ(g)\geq 0$. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive spe…

取值于正性增强算子的归一化势的非交换 Ruelle 定理

A noncommutative Ruelle's Theorem for a normalized potential taking values on positivity-improving operators

本文针对取值于正性增强线性算子的归一化 Lipschitz 势，构造非交换 Ruelle 算子并证明相应 Ruelle 定理，引入 Gibbs 本征态及其熵。

sofic群作用的Lanford-Ruelle定理

The Lanford-Ruelle theorem for actions of sofic groups

本研究将经典Lanford-Ruelle定理推广至sofic群作用情形，证明满足拓扑Markov性质的子移位上，最大化sofic压强的平移不变测度为Gibbs态，还给出平衡测度唯一性等应用及Chung变分原理的简短证明。

Spectral properties of Ruelle transfer operators for regular Gibbs measures and decay of correlations for contact Anosov flows

In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat \cite{D1} and further developed in \cite{St2} is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which pro…

阿贝尔覆盖上的马哈拉姆 - 波利科特 - 吕埃勒共振与自相似平移流

Maharam-Pollicott-Ruelle resonances and self-similar translation flows on abelian covers

研究紧致平移曲面\(\mathbb{Z}^d\)覆盖上自相似平移流关于马哈拉姆测度的遍历性质，基于扭曲转移算子开发重整化方法，得出马哈拉姆 - 波利科特 - 吕埃勒共振，还给出相关应用成果。

随机Ruelle扩张映射的淬火与退火平衡态及其应用

Quenched and annealed equilibrium states for random Ruelle expanding maps and applications

本文研究了波兰空间上扩张映射半群的转移算子谱性质，证明了其存在具有指数相关性衰减和极限定理的平衡态，扩展了基于随机游走驱动的淬火与退火平衡态理论，并展示了在加权非自治迭代函数系统等领域的应用。

The conformal measures of a normal subgroup of a cocompact Fuchsian group

In this paper, we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona's theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations $G$ is hyperbolic then the extremal conformal measures and the hyperbolic boundary of $G$ coincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

周期外力与退化噪声驱动的二维随机 Navier-Stokes 方程的大偏差

Large deviations for 2D stochastic Navier-Stokes Equations driven by a periodic force and a degenerate noise

针对周期力与退化噪声驱动的二维随机 Navier-Stokes 方程，建立时间非齐次 Feynman-Kac 算子的 RPF 定理，并据此证明占据测度的 Donsker-Varadhan 型大偏差原理。

Cones and gauges in complex spaces : Spectral gaps and complex Perron-Frobenius theory

We introduce complex cones and associated projective gauges, generalizing a real Birkhoff cone and its Hilbert metric to complex vector spaces. We deduce a variety of spectral gap theorems in complex Banach spaces. We prove a dominated complex cone-contraction Theorem and use it to extend the classical Perron-Frobenius Theorem to complex matrices, Jentzsch's Theorem to complex integral operators, a Krein-Rutman Theorem to compact and quasi-compact complex operators and a Ruelle-Perron-Frobenius Theorem to complex transfer operators in dynamical systems. In the simplest case of a complex n by n…

带洞拓扑马尔可夫位移的转移算子的拟紧性

Quasi-compactness of transfer operators for topological Markov shifts with holes

本文研究了带2-柱体洞的可数状态拓扑马尔可夫位移的转移算子，在有限不可约转移矩阵和较弱Lipschitz可求和势条件下证明了Ruelle-Perron-Frobenius定理与拟紧性，并计算了逃逸率，还给出了谱隙性质及图迭代函数系统中的应用。

A Ruelle Operator for continuous time Markov Chains

We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $t\geq 0$, taking values on S. We denote by $Ω$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the asso…

转移算子和SRB测度对于Axiom A微分同胚：谱间隙、结构稳定性以及吉布斯等价定理

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。

一般拓扑 Markov 链的若干概率性质

Some Probabilistic Properties of General Topological Markov Chains

本文利用 Markov 过程框架研究转移算子理论，为 Dyson 势相关模型证明了泛函中心极限定理，证明了 Ruelle Operator Formalism 平衡测度的遍历定理，并给出了保形测度满支撑的判定准则。

接触Anosov流的吉布斯测度全部指数混合

Gibbs measures for contact Anosov flows are all exponentially mixing

通过增强Dolgopyat方法，证明接触Anosov流的所有吉布斯测度对Hölder观测具有指数衰减相关性。

On conformal measures and harmonic functions for group extensions

We prove a Perron-Frobenius-Ruelle theorem for group extensions of topological Markov chains based on a construction of $σ$-finite conformal measures and give applications to the construction of harmonic functions.

Axiom A微分同胚的统计极限定理：指数混合、中心极限定理和大偏差

Statistical Limit Theorems for Axiom A Diffeomorphisms: Exponential Mixing, Central Limit Theorem, and Large Deviations

本文研究Axiom A微分同胚的平衡态的统计极限定理，通过Part I中建立的Ruelle转移算子的谱间隙和Part III中的马尔可夫分区编码，证明了体积引理、指数相关衰减、中心极限定理、几乎必然不变原理以及大偏差原理，并展示了这些结果如何从单一的谱机制中推导出来，明确依赖于双曲性数据。

Oscillation and the mean ergodic theorem for uniformly convex Banach spaces

Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p, where the constant C depends only on p and the modulus of uniform convexity. For T a nonexpansive operator, we obtain a weaker bound on the number of epsilon-fluctuations in the sequence. We clarify the relations…

Complex projective metrics on Young towers, countable sub-shifts and other countable covering maps

In this article we will apply complex projective metrics to sequences of complex transfer operators generated by Young towers, countable shifts and other types of distance expanding maps (possibly time dependent) with countable degrees. We will derive a sequential Ruelle-Perron-Frobenius theorem for sequence of complex operators defined by such maps, and relying on it a variety of probabilistic limit theorems (finer than the CLT) for non-stationary processes follows.

On the Asymptotic moments and Edgeworth expansions for some processes in random dynamical environment

We prove that certain asymptotic moments exist for some random distance expanding dynamical systems and Markov chains in random dynamical environment, and compute them in terms of the derivatives at the $0$ of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments $\gam_k=\lim_{n\to\infty}n^{-[\frac k2]}\bbE(\sum_{i=1}^n X_i)^k$ of sums of independent and identically distributed random variables satisfy. We will also obtain certain (Edgeworth) asymptotic expansions related to the central limit theorem for such processes. Our proofs…

非传递随机开、闭动力系统的平衡态

Equilibrium states for non-transitive random open and closed dynamical systems

本文针对非传递随机开、闭区间映射，证明了无传递性条件的随机Ruelle–Perron–Frobenius定理与相对平衡态存在性，拓展了相关理论适用范围并给出多类应用。

Limit theorems for some skew products with mixing base maps

We obtain central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\om,x)=(\te\om,T_\om x)$ together with a $T$-invariant measure, whose base map $\te$ satisfies certain topological and mixing conditions and the maps $T_\om$ on the fibers are certain non-singular distance expanding maps. Our results hold true when $\te$ is either a sufficiently fast mixing Markov shift or a (non-uniform) Young tower with at least one periodic point and polynomial tails. %In fact, our conditions will be satisfied %when $\om$ is the whole orbit t…

Limit theorems for some time dependent expanding dynamical systems

In this paper we will prove various probabilistic limit theorems for some classes of distance expanding sequential dynamical systems (SDS). Our starting point here is certain sequential complex Ruelle-Perron-Frobenius (RPF) theorems which were proved in \cite{book} and \cite{ASIP me} using contraction properties of a complex version of the projective Hilbert metric developed in \cite{Rug}. We will start from the growth rate of the variances of the underlying partial sums. This is well understood in the random dynamics setup, when the maps are stationary, but not in the SDS setup, where various…