Vector valued unified martingale and ergodic theorems with continuous parameter

基于Oseledec乘法遍历定理的矩阵值余弦 finer 过滤

Finer filtration for matrix-valued cocycle based on Oseledec's multiplicative ergodic theorem

本文改进了经典乘法遍历定理，通过更细的过滤方法提升矩阵值余弦分析的精度与稳定性。

A CAT(0)-valued pointwise ergodic theorem

In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space subject to a probability-preserving action of an amenable l.c.s.c. group, where in the CAT(0) setting the role of ergodic averages is played by the barycentres of the empirical distributions of a CAT(0)-valued map along an orbit of the group action. The proof rests on an approximation argument and an appeal to that result for real-valued maps.

算子的一致向量值逐点遍历定理

Uniform vector-valued pointwise ergodic theorems for operators

该研究证明了一类含遍历Koopman算子与压缩乘法算子复合的算子的一致向量值Wiener-Wintner定理，结果在复值函数情形下全新，还适用于部分非正非压缩算子，并给出几类Koopman算子的新一致逐点定理。

Ergodic Properties of weak Asymptotic Pseudotrajectories for Set-valued Dynamical Systems

A successful method to describe the asymptotic behavior of various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes is to relate it to an appropriately chosen limit semiflow. Benaïm and Schreiber (2000) define a general class of such stochastic processes, which they call weak asymptotic pseudotrajectories and study their ergodic behavior. In particular, they prove that the weak* limit points of the empirical measures associated to such processes are almost surely invariant for the associated deterministic semi…

Ergodicity of certain cocycles over certain interval exchanges

We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.

Multifractal analysis of some multiple ergodic averages in linear Cookie-Cutter dynamical systems

In this paper, we study the multiple ergodic averages of a locally constant real-valued function in linear Cookie-Cutter dynamical systems. The multifractal spectrum of these multiple ergodic averages is completely determined.

On the spectrums of ergodic Schrodinger operators with finitely valued potentials

We show that the measure of the spectrum of Schrödinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a quantitative upper bound for the measure of the spectrum. This follows from a result we proved for ergodic Schrödinger operators with finitely valued potentials under two conditions on the recurrence property of the shift. We also show that one of these conditions is necessary for such result.

随机与同步动力学中的乘法遍历定理

A multiplicative ergodic theorem in random and simultaneous dynamics

本文针对紧概率空间上保测度的有限个映射，分别考虑点值路径和集合值路径两种情形，推广了Oseledets乘法遍历定理。

Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups

We consider random fields indexed by finite subsets of an amenable discrete group, taking values in the Banach-space of bounded right-continuous functions. The field is assumed to be equivariant, local, coordinate-wise monotone, and almost additive, with finite range dependence. Using the theory of quasi-tilings we prove an uniform ergodic theorem, more precisely, that averages along a Foelner sequence converge uniformly to a limiting function. Moreover we give explicit error estimates for the approximation in the sup norm.

A Banach space-valued ergodic theorem for amenable groups and applications

In this paper we study unimodular amenable groups. The first part is devoted to results on the existence of uniform families of epsilon-quasi tilings for these groups. In this context, constructions of Ornstein and Weiss are extended by quantitative estimates for the covering properties of the corresponding decompositions. Afterwards, we apply the developed methods to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable, amenable group into some Banach space. This significantly extends and complements the previous results of Lenz, Müller, Schwarzenberger a…

Vector valued $q$-variation for differential operators and semigroups I

In this paper, we establish $\mathcal B$-valued variational inequalities for differential operators, ergodic averages and symmetric diffusion semigroups under the condition that Banach space $\mathcal B$ has martingale cotype property. These results generalize, on the one hand Pisier and Xu's result on the variational inequalities for $\mathcal B$-valued martingales, on the other hand many classical variational inequalities in harmonic analysis and ergodic theory. Moreover, we show that Rademacher cotype $q$ is necessary for the $\mathcal B$-valued $q$-variational inequalities. As applications…

Vector valued $q$-variation for differential operators and semigroups II

In this paper, we establish UMD lattice-valued variational inequalities for differential operators, ergodic averages and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As a consequence, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence.

Uniform ergodic theorems on subshifts over a finite alphabet

We investigate uniform ergodic type theorems for additive and subadditive functions on a subshift over a finite alphabet. We show that every strictly ergodic subshift admits a uniform ergodic theorem for Banach-space-valued additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides in particular a sufficient condition for unique ergodicity.

An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

We study strictly ergodic Delone dynamical systems and prove an ergodic theorem for Banach space valued functions on the associated set of pattern classes. As an application, we prove existence of the integrated density of states in the sense of uniform convergence in distribution for the associated random operators.