We consider volume-preserving flows $(Φ^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(Φ^f_t)_{t\in\mathbb{R}}$ has the form $Φ^f_t(x,y)=(ϕ_tx,y+\int_0^t f(ϕ_sx)ds)$, where $(ϕ_t)_{t\in\mathbb{R}}$ is a locally Hamiltonian flow of hyperbolic periodic type on $S$ and $f$ is a smooth real valued function on $S$. We investigate ergodic properties of these infinite measure-preserving flows and prove that if $f$ belongs to a space of finite codimension in $\mathscr{C}^{2+ε}(S)$, then the following dynamical dichotomy holds: if there is a fixed point of $(ϕ_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, then $(Φ^f_t)_{t\in\mathbb{R}}$ is ergodic, otherwise, if $f$ vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension $(Φ^0_t)_{t\in\mathbb{R}}$. The proof of this result exploits the reduction of $(Φ^f_t)_{t\in\mathbb{R}}$ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of $(ϕ_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.
Finite ergodic index and asymmetry for infinite measure preserving actions
给定$k>0$和一个具有无穷阶元素的阿贝尔可数离散群$G$，我们构造了$(i)$刚性有趣秩1无穷测度保持（i. m. p.）$G$ -遍历指数$k$的作用，$(ii)$ 0-类型有趣的秩一i. m. p. $G$ -遍历指数$k$的作用，$(iii)$有趣的秩一i. m. p. $G$ -遍历指数2的作用$T$，使得乘积$T\times T^{-1}$不是遍历的。证明了对于每一个有趣的秩一$G$ -作用$T$，$T\times T^{-1}$是保守的.
We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension $\hat{T}$ of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.
We show the existence, over an arbitrary infinite ergodic $\mathbb{Z}$-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.
We consider infinite staircase translation surfaces with varying step sizes. For typical step sizes we show that the translation flow is uniquely ergodic in almost every direction. Our result also hold for typical configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology. In contrast, we show that the translation flow on a periodic translation surface can not be uniquely ergodic in any direction.
The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform…
We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are ergodic with isometric coefficients but are not weakly doubly ergodic. We also give type $\text{III}_λ$ examples of such systems, $0<λ\leq 1$. We prove that under certain hypotheses, systems that are weakly mixing are ergodic with isometric coefficients and along the way we give an example of a uniformly rigid topological dynamical system along the sequence $(n_i)…
The ergodic unitarily invariant measures on the space of infinite Hermitian matrices have been classified by Pickrell and Olshanski-Vershik. The much-studied complex inverse Wishart measures form a projective family, thus giving rise to a unitarily invariant measure on infinite positive-definite matrices. In this paper we completely solve the corresponding problem of ergodic decomposition for this measure.
The Ergodic Decomposition of Infinite Pickrell Measures III. The infinite Bessel process as the limit of radial parts of finite-dimensional projections of infinite Pickrell measures
We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti's generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.
Infinite Random Matrices and Ergodic decomposition of Finite or Infinite Hua-Pickrell measures
研究了无限埃尔米特矩阵空间上一类Hua-Pickrell测度族的遍历分解.首先，我们证明了Hua-Pickrell概率测度的遍历分量不含高斯因子，推广了AlexeiBorodin和GrigoriOlshanski的一个结果.其次，我们证明了Hua-Pickrell随机矩阵的渐近特征值序列在一定意义下是平衡的，并且有一个与遍历分量的$γ_1$参数一致的“主值“.这使得我们能够完成Borodin和Olshanski关于Hua-Pickrell概率测度遍历分解的描述。最后，我们将上述结果推广到了无限Hua-Pickrell测度的情形.本文利用A. I. Bufetov，我们能够在$\mathbb{R}^*$中的点配置空间上识别Hua-Pickrell无穷测度到某些显式$σ$ -有限行列式测度的遍历分解.本文解决了Borodin和Olshanski的一个问题。
It is shown that a locally compact second countable group $G$ has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite $σ$-finite standard measure space $(X,μ)$ admitting an exhausting $T$-Følner sequence (i.e. a sequence $(A_n)_{n=1}^\infty$ of measured subsets of finite measure such that $A_1\subset A_2\subset\cdots$, $\bigcup_{n=1}^\infty A_n=X$ and $\lim_{n\to\infty}\sup_{g\in K}\frac{μ(T_gA_n\triangle A_n)}{μ(A_n)}= 0$ for each compact $K\subset G$). It is also shown that a pair of groups $H\sub…
We develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur--Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite er…
