Jon. Aaronson, Tom Meyerovitch
We show that a dissipative, ergodic measure preserving transformation of a sigma-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these.
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We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.
The absolutely continuous spectrum of an ergodic family of one-dimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a number of problems: the absence of absolutely continuous spectrum for rough potentials, the absence of absolute…
Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflections laws. We study the absolutely continuous invariant probabilities of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps having more than one ergodic absolutely continuous invariant probability.
We consider steady states for a class of mechanical systems with particle-disk interactions coupled to two, possibly unequal, heat baths. We show that any steady state that satisfies some natural assumptions is ergodic and absolutely continuous with respect to a Lebesgue-type reference measure and conclude that there exists at most one absolutely continuous steady state.
We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to $0$ in the Cesàro sense. For processes with locally bounded sample paths we show that a spectral functio…
On the tree models of mildly dissipative maps
本文研究圆盘上轻度耗散微分同胚的树模型，关注其兼具遍历非周期数据与原动力学性质，并探讨其作为拓扑及遍历不变量的有效性。
We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.
We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of almost every point. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain a partial extension of Keller's theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly …
We study random perturbations of multidimensional piecewise expanding maps. We characterize absolutely continuous stationary measures (acsm) of randomly perturbed dynamical systems in terms of pseudo-orbits linking the ergodic components of absolutely invariant measures (acim) of the unperturbed system. We focus on those components, called least-elements, which attract pseudo-orbits. We show that each least element admits a neighbourhood which supports exactly one ergodic acsm of the random system. We use this result to identify random perturbations that exhibit a metastable behavior.
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].
Universal Weighted Averaging for Ergodic Flows
本文提出了一种用于遍历流的通用加权平均方法，并通过实例证明了刚性流不具备几乎混合性质。
Deterministic dynamical systems such as the baker maps are useful to shed light on some of the conditions verified by deterministic models in non-equilibrium statistical physics. We investigate a 2D dynamical system, enjoying a weak form of reversibility, with peculiar basins of attraction and steady states. In particular, we test the conditions required for the validity of the Transient Fluctuation Relation. Our analysis illustrates by means of concrete examples why ergodicity of the equilibrium dynamics seems to be a necessary condition for the Transient Fluctuation Relation to hold. This in…
Consider piecewise linear Lorenz maps on $[0, 1]$ of the following form \[ f_{a,b,c}(x)= {ll} ax+1-ac & x \in [0, c) b(x-c) & x \in (c, 1].\] We prove that $f_{a,b,c}$ admits an absolutely continuous invariant probability measure (acim) $μ$ with respect to the Lebesgue measure if and only if $f_{a,b,c}(0) \le f_{a,b,c}(1)$, i.e. $ac+(1-c)b \ge 1$. The acim is unique and ergodic unless $f_{a,b,c}$ is conjugate to a rational rotation. The equivalence between the acim and the Lebesgue measure is also fully investigated via the renormalization theory.
Absolute continuity of stationary measures for random surface dynamics
该研究给出曲面上耗散微分同胚构成的随机动力系统平稳测度绝对连续的条件，覆盖强耗散系统及保体积微分同胚扰动系统，还得到相关轨道分类与等分布结果。
Generic singular continuous spectrum for ergodic Schrödinger operators
考虑具有遍历势$V_ω(n)=f(T^n(ω))$，$n \in \Z$，$ω\in Ω$的薛定谔算子，其中$T:Ω\to Ω$是非周期同胚.我们发现，对于通用的$f \in C(Ω)$，频谱没有绝对连续的组件。证明是基于近似的不连续的潜力，可以通过Kotani理论处理。
We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary ba…
We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^2$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular they hold for Lebesgue almost every rotation interval. The measure obtained is a global physical measure, and it is hyperbolic.
In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center…
We consider a class of dissipative stochastic differential equations (SDE's) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE's to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and for improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions.…
Hyperbolic absolutely continuous invariant measures for C^r one-dimensional maps
本文证明，对 r>1 的 C^r 区间或圆周映射，若在正测度集上 Lyapunov 指数超过阈值，则存在关于 Lebesgue 测度绝对连续的双曲遍历不变测度，且其吸引域几乎处处覆盖该集合。
We study skew product lifts and overlap numbers for equilibrium measures μ_ψof Hölder continuous potentials ψon such lifts. We find computable formulas and estimates for the overlap numbers in several concrete significant cases of systems with overlaps. In particular we obtain iterated systems which are asymptotically irrational-to-1 and absolutely continuous on their limit sets. Then we look into the general structure of the Rokhlin conditional measures of μ_ψwith respect to different fiber partitions associated to the lift Φ, and find relations between them. Moreover we prove an estimate on …
We consider a piecewise smooth expanding map of the interval possessing two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs.
We consider skew products $$T_ϕ:X\times G\to X\times G,~~(x,g)\mapsto(F_1(x),g\;\!ϕ(x)),$$ where $X$ is a compact manifold with probability measure, $G$ a compact Lie group with Lie algebra $\frak g$, $F_1:X\to X$ the time-one map of a measure-preserving flow, and $ϕ\in C^1(X,G)$ a cocycle. Then, we define the degree of $ϕ$ as a suitable function $P_ϕM_ϕ:X\to\frak g$, we show that it transforms in a natural way under Lie group homomorphisms and under the relation of $C^1$-cohomology, and we explain how it generalises previous definitions of degree of a cocycle. For each finite-dimensional irre…
The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map $f$ restricted to its closure is expanding, the property refered to as sub-expanding. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure $m$ with minimal exponent $h$; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a $\sg$--finite invariant measure $μ$ …
An extension of the Wiener-Wintner ergodic theorem for pointwise jointly ergodic systems and its applications
该研究将经典Wiener-Wintner遍历定理推广到带nilsequence权的逐点联合遍历系统，并给出其在加权遍历平均收敛、沿⌊αn⌋子序列遍历平均几乎处处收敛方面的应用。
We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards has a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probability measures (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is always equal to zero. Our main result states that for a generic polygon, if the reflection law has a Lips…
Ergodic properties of rational maps are studied, generalising the work of F.\ Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold, we can construct an induced expanding Markov map with integrable return time which generates the invariant measure.
A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages
本文将van der Corput差分定理推广到Hilbert空间向量序列，借助新Hilbert空间中谱型绝对连续的条件，并应用于非交换保测自同构下的回复性与多重遍历平均问题。
We consider ergodic series of the form $\sum_{n=0}^\infty a_n f(T^n x)$ where $f$ is an integrable function with zero mean value with respect to a $T$-invariant measure $μ$. Under certain conditions on the dynamical system $T$, the invariant measure $μ$ and the function $f$, we prove that the series converges $μ$-almost everywhere if and only if $\sum_{n=0}^\infty |a_n|^2<\infty$, and that in this case the sum of the convergent series is exponentially integrable and satisfies a Khintchine type inequality. We also prove that the system $\{f\circ T^n\}$ is a Riesz system if and only if the spect…
In \cite{J}, Jabłoński proved that a piecewise expanding $C^{2}$ multidimensional Jabłoński map admits an absolutely continuous invariant probability measure (ACIP). In \cite{BL}, Boyarsky and Lou extended this result to the case of i.i.d. compositions of the above maps, with an on average expanding condition. We generalize these results to the (quenched) setting of random Jabłoński maps, where the randomness is governed by an ergodic, invertible and measure preserving transformation. We prove that the skew product associated to this random dynamical system admits a finite number of ergodic AC…
In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota-Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in BGT on admissible random Jabłoński maps to a more general class of higher dimensional random maps.
For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we ca…
We consider stationary ergodic processes indexed by $\mathbb Z$ or $\mathbb Z^n$ whose finite dimensional marginals have laws which are absolutely continuous with respect to Lebesgue measure. We define an entropy theory for these continuous processes, prove an analog of the Shannon Breiman Mac Millan theorem and study more precisely the particular example of Gaussian processes.
We investigate the existence and uniqueness of (locally) absolutely continuous trajectories of a penalty term-based dynamical system associated to a constrained variational inequality expressed as a monotone inclusion problem. Relying on Lyapunov analysis and on the ergodic continuous version of the celebrated Opial Lemma we prove weak ergodic convergence of the orbits to a solution of the constrained variational inequality under investigation. If one of the operators involved satisfies stronger monotonicity properties, then strong convergence of the trajectories can be shown.
We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.
For a sequence of sub-additive potentials, Dai [Optimal state points of the sub-additive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573] gave a method of choosing state points with negative growth rates for an ergodic dynamical system. This paper generalizes Dai's result to the non-ergodic case, and proves that under some mild additional hypothesis, one can choose points with negative growth rates from a positive Lebesgue measure set, even if the system does not preserve any measure that is absolutely continuous with respect to Lebesgue measure.
We study a one-parameter family of interval maps $\{T_α\}_{α\in[1,β]}$, with $β$ the golden mean, defined on $[-1,1]$ by $T_α(x)=β^{1+|t|}x-tβα$ where $t\in\{-1,0,1\}$. For each $T_α,\ α>1$, we construct its unique, absolutely continuous invariant measure and show that on an open, dense subset of parameters $α$, the corresponding density is a step function with finitely many jumps. We give an explicit description of the maximal intervals of parameters on which the density has at most the same number of jumps. A main tool in our analysis is the phenomenon of matching, where the orbits of the le…
We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.
Finitude of physical measures for random maps
本文利用 Markov 算子刻画随机映射的绝对连续遍历平稳测度，建立物理测度有限且吸引盆几乎覆盖全空间的层级与充分条件。
We consider the escaping parameters in the family $β\wp_Λ$, i.e. these parameters for which the orbits of critical values of $β\wp_Λ$ approach infinity, where $\wp_Λ$ is the Weierstrass function. Unlike to the exponential map the considered functions are ergodic. They admit a non-atomic, $σ$-finite, ergodic, conservative and invariant measure $μ$ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on the $\wp_Λ$-function we estimate from below the Hausdorff dimension of the set of escaping parameters in the family $β\wp_Λ$, and compare it with the Hausdorff…
We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and continuous time. The set of all invariant measures is characterized as the feasible set of an infinite-dimensional linear program (LP). The objective functional of this LP is then used to single-out a specific measure (or a class of measures) extremal with respect to the selected functional such as physical measures, ergodic measures, atomic measures (corresponding to, e.g., periodic orbits) or measures absolutely continuous w.r.t. to a gi…
In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov's classical theorem on ergodicity of Anosov diffeomorphisms to the wider class of weakly hyperbolic group actions. To state the second main result, suppose that A denotes a weakly hyperbolic action of a higher rank lattice (such as SL(n,Z) for n>2) by C^2 volume preserving diffeo…
Finitude of physical measures for Markovian random maps
本文研究由遍历马尔可夫链驱动的随机动力系统，利用i.i.d.表示框架建立物理测度有限性条件，并证明在大部分收缩条件下紧度量空间上的随机映射具有有限多个物理测度，且结果可应用于圆周和区间上的随机微分同胚。
We solve the question of the existence of a Poisson-Pinsker factor for conservative ergodic infinite measure preserving action of a countable amenable group by proving the following dichotomy: either it has totally positive Poisson entropy (and is of zero type), or it possesses a Poisson-Pinsker factor. If G is abelian and the entropy positive, the spectrum is absolutely continuous (Lebesgue countable if G=\mathbb{Z}) on the whole L^{2}-space in the first case and in the orthocomplement of the L^{2}-space of the Poisson-Pinsker factor in the second.
Averaging for random metastable systems
本文研究随机扰动下具有两个不变子区间的扩张映射，证明其不变密度可由初始密度的凸组合近似，并确定第二Oseledets空间的极限，推广至多不变集情形。
$M$ is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$, and $f$ is a $C^{1+β}$ ($β>0$) diffeomorphism of $M$. $φ$ is a Hölder continuous potential on $M$. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $φ$), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_χ(p)$, we prove that $φ$ admits a local equilibrium state on $H_χ(p)$ if and only if $φ$ is "recurrent on $H_χ(p)$" (a condition tested by counting periodic points), and one of the leaf measu…
Keplerian shear with Rajchman property
本文研究了开普勒剪切在环 bundle 上的特征，利用拉杰恰曼测度分析了动态系统中测度的衰减速度与拉杰恰曼阶数的关系，并扩展到紧致李群 bundle 情况。
We consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic in the sense that for a fixed base transformation, the set of continuous sampling functions for which the spectral phenomenon occurs is residual. Among the phenomena we discuss are the absence of absolutely continuous spectrum and the vanishing of the Lebesgue measure of the spectrum.
Consider a piecewise smooth expanding map of the interval possessing several invariant subintervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the subintervals lose their invariance in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.
We consider an ergodic invariant measure $μ$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $μ$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element of the action has positive entropy, then $μ$ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.
On systems with finite ergodic degree
本文研究了一类符号动力系统$(Ø, T, μ)$的遍历理论，其中$T:Ø\to Ø$是$Ø=\prod_0^\infty\{0,1\}$上的左移变换，$μ$是一个$\s$ -有限$T$ -不变测度，它具有这样的性质：可以找到一个实数$d$，使得对所有$ε>0$，$μ(τ^d)=\infty$，但$μ(τ^{d-ε})<\infty$，其中$τ$是参考状态1中的第一个通过时间函数.特别是，我们将考虑不变措施$μ$所产生的潜在的$V$是一致连续的，但不是可求和的变化。如果$d>0$，则$μ$可以被归一化以给出$V$的唯一非原子平衡概率测度，我们计算（渐近）精确混合率，阶为$n^{-d}$。我们还建立了弱伯努利属性和多项式簇属性（衰减的相关性）的观测多项式变化。如果改为$d\leq 0$，则$μ$是具有$n^d$阶缩放率的无限测度。此外，加权动态zeta函数和相关函数的傅立叶变换的分析性质示出为通过一个运营商值的幂级数，自然产生的标准诱导过程的频谱特性相互关联。这些功能的奇异行为在其附近的非极性奇异$z=1$的详细控制是通过一个近似方案，它使用一个合适的更新过程的生成函数。从可微动力学的观点出发，给出了一类分段光滑区间映射具有中立不动点的唯一绝对连续不变测度。
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