Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem
英文摘要:
A very short and direct proof along the lines of the Kamae-Katznelson-Weiss approach.
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。
Karl Petersen
A very short and direct proof along the lines of the Kamae-Katznelson-Weiss approach.
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
We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.
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。
We give a short combinatorial proof of the classical pointwise ergodic theorem for probability measure preserving $\mathbb{Z}$-actions. Our approach reduces the theorem to a tiling problem: tightly tile each orbit by intervals with desired averages. This tiling problem is easy to solve for $\mathbb{Z}$ with intervals as tiles. However, it would be interesting to find other classes of groups and sequences of tiles for which this can be done, since then our approach would yield a pointwise ergodic theorem for such classes.
A classification of intrinsic ergodicity for recognisable random substitution systems
本文研究了由随机替换生成的动力系统,探讨了内在ergodic性和多最大熵测度的分类,证明了最大熵测度通过适当的对称关系不变性分类,并给出了内在ergodic性的充分条件。
A theorem due to Hindman states that if $E$ is a subset of $\mathbb{N}$ with $d^*(E)>0$, where $d^*$ denotes the upper Banach density, then for any $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that $d^*\left(\bigcup_{i=1}^N(E-i)\right) > 1-\varepsilon$. Curiously, this result does not hold if one replaces the upper Banach density $d^*$ with the upper density $\bar{d}$. Originally proved combinatorially, Hindman's theorem allows for a quick and easy proof using an ergodic version of Furstenberg's correspondence principle. In this paper, we establish a variant of the ergodic Furstenberg'…
Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously
本文提出通用投影定理,在横截性和相对维数可分条件下证明投影维数等式及近似双 Lipschitz 性质对一族测度同时成立,并应用于多分形分析、遍历测度和 Gibbs 测度的维数公式。
Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences
本文研究了Hardy序列的半范数估计与成对独立Hardy序列的联合遍历性,推导了多维模式的复发结果并解决了联合遍历性分类问题。
In this paper, we establish the one-sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative $L_p$-spaces for a fixed $1<p<\infty$, which particularly applies to positive isometries and general positive Lamperti contractions or power bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither the category of positive contractions considered by Junge-Xu nor the class of power bounded positive invertible operators considered by Hong…
Ergodic averages for sparse sequences along primes
本文研究了素数处稀疏序列的多重遍历平均极限行为,利用Gowers一致性、提升论证与多项式逼近等方法,证明了正密度集合包含步长为$\lfloor p^c \rfloor$的等差数列,确立了Frantzikinakis猜想。
B. Green and T. Tao have recently proved that 'the set of primes contains arbitrary long arithmetic progressions', answering to an old question with a remarkably simple formulation. The proof does not use any "transcendental" method and any of the deep theorems of analytic number theory. It is written in a 'spirit' close to ergodic theory and in particular of Furstenberg's proof of Szemerédi's Theorem, but it does not use any result of this theory. Therefore the method can be considered as elementary, which does not mean easy. We entend here to present the mains ideas of this proof.
Let $μ$ be a probability measure on $\mathbb{R}/\mathbb{Z}$ that is ergodic under the $\times p$ map, with positive entropy. In 1995, Host showed that if $\gcd(m,p)=1$ then $μ$ almost every point is normal in base $m$. In 2001, Lindenstrauss showed that the conclusion holds under the weaker assumption that $p$ does not divide any power of $m$. In 2015, Hochman and Shmerkin showed that this holds in the "correct" generality, i.e. if $m$ and $p$ are independent. We prove a simultaneous version of this result: for $μ$ typical $x$, if $m>p$ are independent, we show that the orbit of $(x,x)$ under …
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