Van Cyr, Bryna Kra
Using results relating the complexity of a two dimensional subshift to its periodicity, we obtain an application to the well-known conjecture of Furstenberg on a Borel probability measure on $[0,1)$ which is invariant under both $x\mapsto px \pmod 1$ and $x\mapsto qx \pmod 1$, showing that any potential counterexample has a nontrivial lower bound on its complexity.
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Polynomial ergodic averages of measure-preserving systems acted by $\mathbb{Z}^{d}$
该研究通过把一般 $\mathbb{Z}^{d}$ 作用保测系统的多项式遍历平均逐点收敛归结为零熵情形，证明了 $\mathbb{Z}^{d}$ 作用 K-系统中的相应收敛性。
Amos Nevo established the pointwise ergodic theorem in $L^p$ for measure-preserving actions of $\mathrm{PSL}_2(\mathbb{R})$ on probability spaces with respect to ball averages and every $p>1$. This paper shows by explicit example that Nevo's Theorem cannot be extended to $p=1$.
It is well-known that any $\mathbb{Z}$ subshift with the specification property has the property that every factor is intrinsically ergodic, i.e., every factor has a unique factor of maximal entropy. In recent work, other $\mathbb{Z}$ subshifts have been shown to possess this property as well, including $β$-shifts and a class of $S$-gap shifts. We give two results that show that the situation for $\mathbb{Z}^d$ subshifts with $d >1 $ is quite different. First, for any $d>1$, we show that any $\mathbb{Z}^d$ subshift possessing a certain mixing property must have a factor with positive entropy w…
In this paper we show that the natural action of the symmetric group acting on the product space $\{0, 1 \}^{\mathbb{N}} $ endowed with a symmetric measure is approximately transitive. We also extend the result to a larger class of probability measures.
Let $m\in\mathbb{N}$ and $\textbf{X}=(X,\mathcal{X},μ,(T_α)_{α\in\mathbb{R}^{m}})$ be a measure preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $ν$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\textbf{X}$ if there exists $A\subseteq\mathbb{R}$ of density 1 such that for all $f\in L^{\infty}(μ)$, we have $$\lim_{t\in A,t\to\infty}\int_{\mathbb{R}^{m}}f(T_{t α}x)\,dν(α)=\int_{X}f\,dμ$$ for $μ$-a.e. $x\in X$. Let $W(\textbf{X})$ denote the collection of all $α\in\mathbb{R}^{m}$ such that the $\mathbb{R}$-action $(T_{tα})_{t\in\mathbb{R}}$ is not ergodic. Under…
Ergodic measures of intermediate entropies for $\mathbb{Z}^{d}$-action
本文研究满足近似$\mathbb{Z}^{d}$或$\mathbb{Z}_+^{d}$-积性质且渐近熵扩张的动力系统，证明了其不变测度空间的结构，并证明了具有任意给定中间熵的遍历测度在某些自然子空间中是稠密的，从而验证了Katok关于此类系统存在任意中间熵遍历测度的猜想。
We adapt techniques of Hochman to prove a non-singular ergodic theorem for $\mathbb{Z}^d$-actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for a class of product actions.
The non-ergodic Host-Kra-Ziegler structure theorem for $\mathbb{Z}^d$-actions via measurable selections
通过可测选择程序将非遍历情形归约到遍历定理，建立了保测$\mathbb{Z}^d$作用的非遍历Host-Kra-Ziegler结构定理，并给出了垂直零字符版本，该定理是第二作者与Fraczyk分类$\mathrm{ASL}_d(\mathbb{Z})$不变点过程的关键输入。
A subshift with linear block complexity has at most countably many ergodic measures, and we continue of the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but has uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an ergodic measure whose slow entropy grows slower than any given rate tending to infinitely but faster than any other rate m…
We study the space of ergodic measures of the map $$f:\mathbb{T}^2\to \mathbb{T}^2, \ f(x, y)=(x, \ x+y)(\text{mod}\, 1),$$ and show that its structure is similar to the graph of Thomae's function.
Quantitative expansivity for ergodic $\mathbb{Z}^d$ actions
本文研究了遍历$\mathbb{Z}^d$作用在循环子群和整系数多项式像上的定量扩张性，统一并强化了已有方法，并意外为多项式Bogolyubov定理的特定固定变体提供了反例。
A Borel system $(X,S)$ is `almost Borel universal' if any free Borel dynamical system $(Y,T)$ of strictly lower entropy is isomorphic to a Borel subsystem of $(X,S)$, after removing a null set. We obtain and exploit a new sufficient condition for a topological dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a `generic' homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non-uniform specification impli…
We prove that the ergodic deviation of a degenerate $\mathbb{Z}^2$-action on the torus $\mathbb{T}^2$ relative to a symmetric, strictly convex body can be decomposed into two parts, and that each part admits a limit distribution after choosing a suitable normalizer. Specifically, the first part is similar to an ergodic sum of smooth observables after being normalized by $N$, and the second part is similar to the case of a random toral translation, i.e., the $\mathbb{Z}$-action, but with a normalizer of $N^{\frac{1}{2}}$. The key difference is that we employ the product flow on the product spac…
We define a random walk adic transformation associated to an aperiodic random walk on $G=\mathbb{Z}^{k}\times\mathbb{R}^{D-k}$ driven by a $β$-transformation and study its ergodic properties. In particular, this transformation is conservative, ergodic, infinite measure preserving and we prove that it is asymptotically distributionally stable and bounded rationally ergodic. Related earlier work appears in [AS] and [ANSS] for random walk adic transformations associated to an aperiodic random walk driven by a subshift of finite type.
In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{Hölder}}$ diffeomorphisms of compact manifolds. We construct intermediate and coarse unstable manifolds for the action and establish controls on their local geometry. In the second part we consider the relationship between entropy, Lyapunov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier--You…
Let $K$ be a compact metrizable group and $\Ga$ be a finitely generated group of commuting automorphisms of $K$. We show that ergodicity of $\Ga$ implies $\Ga$ contains ergodic automorphisms if center of the action, $Z(\Ga) = \{\ap \in {\rm Aut}(K) \mid \ap {\rm commutes with elements of \rm} \Ga \}$ has DCC. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which in particular, retrieves Theorems of Berend \cite{Be} and Schmidt \cite{Sc1} proved in this context.
Weighted multiple ergodic averages via analytic observables over $ \mathbb{T}^\infty $: Is exponential pointwise convergence universal?
本文通过加权方法研究了在无限维环上多重遍历平均的指数点wise收敛性，解决了其普遍性问题，并通过反例和数值分析验证了方法的最优性。
In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length n relates to the one for words of length …
Let $(X,\mathcal{B},μ)$ be a probability space and let $T_1,..., T_l$ be $l$ commuting invertible measure preserving transformations \linebreak of $X$. We show that if $T_1^{c_1} ... T_l^{c_l}$ is ergodic for each $(c_1,...,c_l)\neq (0,...,0)$, then the averages $\frac{1}{|Φ_N|}\sum_{u\inΦ_N}\prod_{i=1}^r T_1^{p_{i1}(u)}... T_l^{p_{il}(u)}f_i$ converge in $L^2(μ)$ for all polynomials $p_{ij}\colon \mathbb{Z}^d\to\mathbb{Z}$, all $f_i\in L^{\infty}(μ)$, and all Følner sequences $\{Φ_N\}_{N=1}^{\infty}$ in $\mathbb{Z}^d$.
Entropy and strict ergodicity for multidimensional Toeplitz subshifts
本研究通过自洽构造，证明了具有任意规定熵的严格遍历Toeplitz Z^d-子移位存在，且其极大等连续因子均为Z^d上的通用里程计。
In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are neglected in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous dynamical systems or continuous observation functions. Meanwhile, on the torus $\mathbb{T}^{d}$ with special rotation, we prove the pointwise convergence of multiple ergodic average $\disp \f 1 N \sum_…
We show that there are an irrational rotation $Tx=x+α$ on the circle $\mathbb{T}$ and a continuous $φ\colon\mathbb{T}\to\mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{φ,\mathcal{S}}$ acting on $(X\times Y,μ\otimesν)$ by the formula $T_{φ,\mathcal{S}}(x,y)=(Tx,S_{φ(x)}(y))$, where $μ$ stands for Lebesgue measure on $\mathbb{T}$ and $ν$ denotes the unique $\mathcal{S}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mi…
LLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $ν$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h\_ν> \log d$ have a directional dimension $>2$, which answers a question by de Thélin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Lattès maps. Finally, we obtain an upper bound for the dimension of the eq…
For any 1-lipschitz ergodic map $F:\; \mathbb{Z}^{k}_{p} \mapsto \mathbb{Z}^{k}_{p},\;k>1\in\mathbb{N},$ there are 1-lipschitz ergodic map $G:\; \mathbb{Z}_{p} \mapsto \mathbb{Z}_{p}$ and two bijection $H_k$, $T_{k,\;P}$ that $$G = H_{k} \circ T_{k,\;P}\circ F\circ H^{-1}_{k} \text{ and } F = H^{-1}_{k} \circ T_{k,\;P^{-1}}\circ G\circ H_{k}.$$
Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of $\mathbb{F}_{p}^ω$. From this we deduce multiple Khintchine-type recurrence results analogous to those for $\mathbb{Z}$-systems obtained by Bergelson, Host, and Kra, and also present some new counterexamples in this setting.
Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity
本文研究一类含滑动Shilnikov连接附近首次回归映射的一维分段扩张映射，借助共形迭代函数系统理论，证明其不变集上动力学与$\be{N}^{\be{N}}$上移位拓扑共轭，存在唯一不变遍历共形测度。
In this paper we consider invertible one-dimensional linear cellular automata (CA hereafter) defined on a finite alphabet of cardinality $p^k$, i.e. the maps $T_{f[l,r]}:\mathbb{Z}^{\mathbb{Z}}_{p^k}\to\mathbb{Z}^{\mathbb{Z}}_{p^k}$ which are given by $T_{f[l,r]}(x) = (y_n)_{n=-\infty}^{\infty} $, $y_{n} = f(x_{n+l}, ..., x_{n+r}) =\overset{r}{\underset{i=l}{\sum}}λ_{i}x_{n+i}(\text{mod} p^k)$, $x=(x_n)_{n=-\infty}^{\infty}\in \mathbb{Z}^{\mathbb{Z}}_{p^k}$ and $f:\mathbb{Z}^{r-l+1}_{p^k}\to \mathbb{Z}_{p^k}$, over the ring $\mathbb{Z}_{p^k}$ $(k \geq 2$ and $p$ is a prime number), where $gcd(…
On the rate of convergence of $\mathbb{R}^d$-ergodic averages constructed over a strictly convex set
本文针对严格凸集上构造的 $\mathbb{R}^d$ 遍历平均，利用谱判据和 Hertz 的傅里叶渐近结果，证明其收敛速率与集合具体形状无关。
Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations
本文研究了区间交换变换上周期型$\mathbb{Z}^m$-斜积的遍历不变Radon测度分类问题，通过引入adic编码的符号语言并应用相关符号结果，证明了此类测度恰为由$\mathbb{R}^m$参数化的Maharam测度，并给出了其显式形式与弱*-连续性。
Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological dynamical systems, and in asymptotics for patterns in certain subsets of the integers. We show, however, that many natural classes in both measure preserving systems and topological dynamical systems contain no higher order nilsystems as factors, meaning that the only nilsystems they contain as factors are rotations. In the ergodic setting, we show that there are spectral obstructions that give rise …
For an ergodic hyperbolic measure $ω$ of a $C^{1+α}$ diffeomorphism, there is an $ω$ full-measured set $\tildeΛ$ such that every nonempty, compact and connected subset $V$ of $\mathbb{M}_{inv}(\tildeΛ)$ coincides with the accumulating set of time averages of Dirac measures supported at {\it one orbit}, where $\mathbb{M}_{inv}(\tildeΛ)$ denotes the space of invariant measures supported on $\tildeΛ$. Such state points corresponding to a fixed $V$ are dense in the support $supp(ω)$. Moreover, $\mathbb{M}_{inv}(\tildeΛ)$ can be accumulated by time averages of Dirac measures supported at {\it one o…
Let $\mathcal{P}$ be an (unbounded) countable multiset of primes, let $G=\bigoplus_{p\in P}\mathbb{F}_p$. We study the $k$'th universal characteristic factors of an ergodic probability system $(X,\mathcal{B},μ)$ with respect to some measure preserving action of $G$. We find conditions under which every extension of these factors is generated by phase polynomials and we give an example of an ergodic $G$-system that is not Abramov. In particular we generalize the main results of Bergelson Tao and Ziegler who proved a similar theorem in the special case $P=\{p,p,p,...\}$ for some fixed prime $p$.…
The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,\mathcal{B},μ,T)$ and set $A \in \mathcal{B}$ with $μ(A) > 0$, one has $c(A):= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N μ(A \cap T^{-P(n)}A) > 0.$ The limit $c(A)$…
A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental propert…
It is well-known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynam…
We prove that given a measure preserving system $(X,\mathcal{B},μ,T_1,\dots,T_d)$ with commuting, ergodic transformations $T_i$ such that $T_iT_j^{-1}$ are ergodic for all $i \neq j$, the multicorrelation sequence $a(n)=\int_X f_0 \cdot T_1^nf_1 \cdot \dotso \cdot T_d^n f_d \ dμ$ can be decomposed as $a(n)=a_{\textrm{st}}(n)+a_{\textrm{er}}(n)$, where $a_{\textrm{st}}$ is a uniform limit of $d$-step nilsequences and $a_{\textrm{er}}$ is a nullsequence (that is, $\lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |a_{\textrm{er}}|^2=0$). Under some additional ergodicity conditions on $T_1,\do…
We introduce computable actions of computable groups and prove the following versions of effective Birkhoff's ergodic theorem. Let $Γ$ be a computable amenable group, then there always exists a canonically computable tempered two-sided Følner sequence $(F_n)_{n \geq 1}$ in $Γ$. For a computable, measure-preserving, ergodic action of $Γ$ on a Cantor space $\{0,1\}^{\mathbb N}$ endowed with a computable probability measure $μ$, it is shown that for every bounded lower semicomputable function $f$ on $\{0,1\}^{\mathbb N}$ and for every Martin-Löf random $ω\in \{0,1\}^{\mathbb N}$ the equality \[ \…
We introduce the fundamental group ${\mathcal F}(\mathcal{R}_{G, ϕ})$ of a uniquely ergodic Cantor minimal $G$-system $\mathcal{R}_{G, ϕ}$ where $G$ is a countable discrete group. We compute fundamental groups of several uniquely ergodic Cantor minimal $G$-systems. We show that if $\mathcal{R}_{G, ϕ}$ arises from a free action $ϕ$ of a finitely generated abelian group, then there exists a unital countable subring $R$ of $\mathbb{R}$ such that $\mathcal{F}(\mathcal{R}_{G, ϕ})=R_{+}^\times$. We also consider the relation between fundamental groups of uniquely ergodic Cantor minimal $\mathbb{Z}^n…
Disintegrations of non-hyperbolic ergodic measures along the center foliation of DA maps
该研究针对3维环面上同伦于Anosov的部分双曲微分同胚，证明其非双曲遍历测度存在与每个中心叶交点至多两个的满测度子集，推进了非双曲遍历测度沿中心叶层分解的研究。
We study certain multiple ergodic averages of an iterated functions system generated by two contractions on the unit interval. By using the dynamical coding ${0,1}^{\mathbb{N}}$ of the attractor, we compute the Hausdorff dimension of the set of points with a given frequency of the pattern 11 in positions $k, 2k$.
Pointwise convergence of some continuous-time polynomial ergodic averages
本文基于可测流的拓扑模型研究连续时间多项式遍历平均的逐点收敛性，证明了特定条件下相关极限对几乎处处点存在，并建立了涉及测地流和水平圆周流的逐点遍历定理。
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 …
Fluctuation of ergodic averages and other stochastic processes
本文证明对固定遍历映射，通有的非平凡L¹函数的遍历平均几乎处处围绕均值无限次波动，并将该通有波动性推广到遍历平均子序列、卷积算子、加权平均、均匀分布和鞅等随机过程。
It was shown by S. Kalikow and B. Weiss that, given a measure-preserving action of $\mathbb{Z}^d$ on a probability space $X$ and a nonnegative measurable function $f$ on $X$, the probability that the sequence of ergodic averages $$ \frac 1 {(2k+1)^d} \sum\limits_{g \in [-k,\dots,k]^d} f(g \cdot x) $$ has at least $n$ fluctuations across an interval $(α,β)$ can be bounded from above by $c_1 c_2^n$ for some universal constants $c_1 \in \mathbb{R}$ and $c_2 \in (0,1)$, which depend only on $d,α,β$. The purpose of this article is to generalize this result to measure-preserving actions of groups of…
Let $O(\infty)$ and $U(\infty)$ be the inductively compact infinite orthogonal group and infinite unitary group respectively. The classifications of ergodic probability measures with respect to the natural group action of $O(\infty)\times O(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{R})$ and that of $U(\infty)\times U(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{C})$ are due to Olshanski. The original proofs for these results are based on the asymptotic representation theory. In this note, by applying the Vershik-Kerov method, we propose a simple method for obtaining these two classi…
Starting from a discrete $C^*$-dynamical system $(\mathfrak{A}, θ, ω_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(\mathfrak{A}\rtimes_α\mathbb{Z}, Φ_{θ, u},\om_o\circ E)$, $E:\mathfrak{A}\rtimes_α\mathbb{Z}\rightarrow\ga$ being the canonical conditional expectation of $\mathfrak{A}\rtimes_α\mathbb{Z}$ onto $\mathfrak{A}$, provided $\a\in\aut(\ga)$ commute with the $*$-automorphism $þ$ up tu a unitary $u\in\ga$. Here, $Φ_{θ, u}\in\aut(\mathfrak{A}\rtimes_α\mathbb{Z})$ can be considered as the fully noncommutative generalisation of …
Rate distortion dimension and ergodic decomposition for $\mathbb{R}^d$-actions
本文研究R^d-作用的率失真维数在遍历分解下的行为，证明了在不变概率测度凸组合下上、下率失真维数分别具有凸性与凹性，并给出例子说明定理的适用边界。
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.
Ergodic Averages over Shrinking Sectors of $\mathbb Z[i]$
本文研究高斯整数收缩角扇形上的遍历平均，证明唯一遍历系统中连续函数平均值一致收敛，并给出有限生成乘法作用的定量比较，推论包括角位置独立性、高斯刘维尔相消及素数因子计数的联合等分布。
On the non-commutative Neveu decomposition and stochastic ergodic theorems
该研究证明了半有限及σ-有限von Neumann代数上顺从半群/群作用的Neveu分解，并得到非交换L¹空间上正压缩的随机遍历定理，填补了Danford-Schwartz范畴外的相关空白。
Ergodic $\times p$-invariant measures on $\mathbb{T}^2$ with no dimension dropping projections
本研究构造了二维环面上维数为$s$的遍历$\times p$-不变测度，其所有直线投影均保持维数且平面管质量满足$O(w^s)$界，同时证明正维数拟Bernoulli $T_p$-不变测度必存在维数下降投影。
An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic $μ$-invariant for Kubota--Leopoldt $p$-adic $L$-functions is an equidistribution result which they established using the Weyl criterion. The purpose of our manuscript is to provide an alternative proof by adopting a dynamical approach. A key ingredient to our methods is studying an ergodic skew-product map on $\mathbb{Z}_p \times [0,1]$, which is then suitably identified as a factor of the $2$-sided Bernoulli shift on the sample space $\{0,1,2,\cdots,p-1\}^{\mathbb{Z}}$.
We study double averages along orbits for measure preserving actions of $\mathbb{A}^ω$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation estimate for these averages, which in particular reproves their convergence in $L^p$ for any finite $p$ and for any choice of two $L^\infty$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.
Irreducible groups and ergodicity in the boundary
该研究证明了在特定条件下，离散子群无法与自由群或双曲群等价，并探讨了李群的代数性质与遍历性之间的关系。
We prove a.e. convergence of continuous-time quadratic averages with respect to two commuting $\mathbb{R}$-actions, coming from a single jointly measurable measure-preserving $\mathbb{R}^2$-action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.
Let $G$ be a countable abelian group. We study ergodic averages associated with configurations of the form $\{ag,bg,(a+b)g\}$ for some $a,b\in\mathbb{Z}$. Under some assumptions on $G$, we prove that the universal characteristic factor for these averages is a factor of a $2$-step nilpotent homogeneous space. As an application we derive a Khintchine type recurrence result. In particular, we prove that for every countable abelian group $G$, if $a,b\in\mathbb{Z}$ are such that $aG,bG,(b-a)G$ and $(a+b)G$ are of finite index in $G$, then for every $E\subset G$ and $\varepsilon>0$ the set $$\{g\in …
For $τ>2$, let $T$ be a $C^τ$ skew product map of the form $(x+α,y+h(x))$ on $\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the Möbius sequence. This in particular implies that any non-uniquely ergodic $C^τ$ skew product map on $\mathbb T^2$ has a finite index factor that is disjoint to the Möbius sequence.
We study the Kronecker sequence $\{nα\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $α$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.
Let $F$ be a non-discrete non-Archimedean locally compact field such that the characteristic $\mathrm{ch}(F)\ne 2$ and let $\mathcal{O}_F$ be the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrm{Skew}(\mathbb{N}, F)$ of infinite skew-symmetric matrices with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F)$ and Theorem 1.4, that gives an unexpected natural correspondence between the set of $\mathrm{GL}(\infty,\mathcal{O}_F)$-invariant Borel probability measures on $\mathrm{Sym}…
Hitting times of shrinking targets: Transversality and an ergodic theorem
本文研究收缩目标集的命中时间，证明其遍历平均点态收敛性及与任意固定集合的几何独立性，通过整数-分形维数刻画。
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