Idris Assani, Zoltan Buczolich, Daniel Mauldin
We solve the following counting problem for measure preserving transformations. For $f\in L_+^1(μ)$, is it true that $\ds \sup_n\frac{\bN_n(f)(x)}{n} <\infty,$ where $$\ds\bN_n(f)(x)= # {k: \frac{f(T^k x)}{k}>\frac 1 n}?$$ One of the consequences is the nonvalidity of J. Bourgain's Return Time Theorem for pairs of $(L^1, L^1)$ functions.
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In her thesis, Mirzakhani showed that the number of simple closed geodesics of length $\leq L$ on a closed, connected, oriented hyperbolic surface $X$ of genus $g$ is asymptotic to $L^{6g-6}$ times a constant depending on the geometry of $X$. In this survey we give a detailed account of Mirzakhani's proof of this result aimed at non-experts. We draw inspiration from classic primitive lattice point counting results in homogeneous dynamics. The focus is on understanding how the general principles that drive the proof in the case of lattices also apply in the setting of hyperbolic surfaces.
For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications are found in an affine sieve on orbits of thin groups as well as in sphere counting problems for sphere packings invariant under a geometrically finite group. In our sphere counting problems, spheres can be ordered with respect to a general conformal metric.
We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},ν,τ)$ and any $f\in L^p(ν)$, $p>1$, the limit $$ \lim_{n\to \infty}{1\over \sum_{k=1}^{n} d(k)} \sum_{k=1}^{n} d(k)f(τ^k x)$$ exists $ν$-almost everywhere. We also obtain similar results for other arithmetical functions, like $θ(n)$ function counting the number of squarefree divisors of $n$ and the generalized Euler totient function $J_s(n)$, $s>0$. We use Bourgain's method, namely the…
Let $M=X/Γ$ be a geometrically finite negatively curved manifold with fundamental group $Γ$ acting on $X$ by isometries. The purpose of this paper is to study the mixing property of the geodesic flow on $T^1M$, the asymptotic equivalent as $R\longrightarrow+\infty$ of the number of closed geodesics on $M$ of length less than $R$ and of the orbital counting function $\sharp\{γ\inΓ |\ d(o,γ.o)\le R\}$. These properties are well known when the Bowen-Margulis measure on $T^1M$ is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise…
We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Quantitative ergodic theorems give rise to new and previously inaccessible applications, and we demonstrate the remarkable diversity of such applications by presenting several nu…
In 1984 Boshernitzan proved an upper bound on the number of ergodic measures for a minimal subshift of linear block growth and asked if it could be lowered without further assumptions on the shift. We answer this question, showing that Boshernitzan's bound is sharp. We further prove that the same bound holds for the, a priori, larger set of nonatomic generic measures, and that this bound remains valid even if one drops the assumption of minimality. Applying these results to interval exchange transformations, we give an upper bound on the number of nonatomic generic measures of a minimal IET, a…
The article presents a new perspective on the isomorphism problem for non-ergodic measure-preserving dynamical systems with discrete spectrum which is based on the connection between ergodic theory and topological dynamics constituted by topological models. By first solving the isomorphism problem for a certain class of topological dynamical systems, it is shown that the measure-preserving case can in fact be deduced from the topological one via the construction of topological models. As a byproduct, a new characterization of mean ergodicity for topological dynamical systems is obtained.
Given a finite-to-one factor map $π: (X, T) \to (Y, S)$ between topological dynamical systems, we look into the pushforward map $π_*: M(X, T) \to M(Y,T)$ between sets of invariant measures. We investigate the structure of the measure fiber $π_*^{-1}(ν)$ for an arbitrary ergodic measure $ν$ on the factor system $Y$. We define the degree $d_{π,ν}$ of the factor map $π$ relative to $ν$ and the multiplicity of each ergodic measure $μ$ on $X$ that projects to $ν$, and show that the number of ergodic pre-images of $ν$ is $d_{π,ν}$ counting multiplicity. In other words, the degree $d_{π,ν}$ is the su…
We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d > 1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project se…
Resolving the joint ergodicity problem for Hardy sequences
该研究针对多项式增长Hardy序列，在无独立性假设下解决联合遍历性分类问题的困难方向，发现逆命题对病态族不成立，还提出了若干有独立价值的新理论工具。
We propose to study the multifractal behavior of weighted ergodic averages. Our study in this paper is concentrated on the symbolic dynamics. We introduce a thermodynamical formalism which leads to a multifractal spectrum. It is proved that this thermodynamical formalism applies to different kinds of dynamically defined weights, including stationary ergodic random weights, uniquely ergodic weights etc. But the validity of the thermodynamical formalism for very irregular weights, like Möbius function, is an unsolved problem. The paper ends with some other unsolved problems.
The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n \in \mathbb{Z}$ such that $T^n A \cap A \not = \varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure $\{T_i\}$, there exists a rank-one transformation $S$ such that $T_i \times S$ is not ergodic for all $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $\mathbb{Z}^d$ actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ …
Minimax aspects of optimizations in ergodic theory
该研究从极小极大视角探究遍历理论优化问题，给出最大遍历平均的极小极大刻画，其方法适用于广义压力函数的抽象变分原理，并揭示了与Fenchel-Rockafellar对偶性的关联。
We study the error of the number of unimodular lattice points that fall into a dilated and translated parallelogram. By using an article from Skriganov, we see that this error can be compared to an ergodic sum that involves the discrete geodesic flow over the space of unimodular lattices. With the right normalization, we show, by using tools from a previous work of Fayad and Dolgopyat, that a certain point process converges in law towards a Poisson process and deduce that the ergodic sum converges towards a Cauchy centered law when the unimodular lattice is distributed according to the normali…
We show that for an expanding map, the maximizing measures of a generic (open and dense) $C^r$ ($r\in\mathbb{N}$) differentiable functions are supported on a single periodic orbit. [There is a gap in the discussions. For the $C^{\infty}$ approximation of the Lipschitz functions, we can only control the $C^1$ derivative, but we can not control the $C^r$ derivatives for $r\geq2$. Elegant approximation methods might be needed to solve this problem.]
On orthogonality to uniquely ergodic systems
本文利用Furstenberg系统解决了Boshernitzan关于有界序列与所有唯一遍历系统正交性的刻画问题，并进一步用平均Chowla性质刻画了与离散谱唯一遍历系统正交的序列。
We formulate a general statement of the problem of defining invariant measures with certain properties and suggest an ergodic method of perturbations for describing such measures.
We apply lattice points counting results of Gorodnik and Nevo to solve a shrinking target problem in the setting of geodesic flows on hyperbolic manifolds of finite volume.
Counting Problem for Some Random Conformal Iterated Function Systems
本文研究随机共形迭代函数系统的计数问题，发现随机情形下计数性质与确定性系统根本不同，非指数增长可在满测度集上构造，且随机游走的常返行为起关键作用。
In this paper we present a complete solution to the problem of multifractal analysis of multiple ergodic averages in the case of symbolic dynamics for functions of two variables depending on the first coordinate.
Multiple mixing in ergodic theory
本文回顾并讨论Rokhlin于1949年提出的k重混合及相关度量不变量不匹配这一遍历理论长期开放问题。
This work contains the following results: the trajectory fullness of the homoclinic groups, their connections with factors, K-property, weak multiple mixing; the ergodicity of the weakly homoclinic group for Gauss and Poisson actions; the triviality of the weakly homoclinic group for classes of rank-1 actions. Some open problems are discussed.
Ergodic optimization is the study of problems relating to maximizing orbits, maximizing invariant measures and maximum ergodic averages. An orbit of a dynamical system is called f-maximizing if the time average of the real-valued function f along the orbit is larger than along all other orbits, and an invariant probability measure is called f-maximizing if it gives f a larger space average than does any other invariant probability measure. In this survey we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimiza…
We survey some recent developments and give a list of open problems regarding multiple recurrence and convergence phenomena of $\mathbb{Z}^d$ actions in ergodic theory and related applications in combinatorics and number theory.
Representations of binary quadratic forms by quaternary quadratic forms
本文通过Linnik的ergodic方法变种，证明了二元二次型由四元二次型的原始表示的局部-全局原理，并利用Einsiedler和Lindenstrauss的测度分类结果和Siegel质量公式，解决了一个affine variety上的计数问题。
A generalization of the regular continued fractions was given by Chakraborty and Rao \cite{CR-2003}. For the transformation which generates this expansion and its invariant measure, the Perron-Frobenius operator is given and studied. For this expansion, we apply the method of Rockett and Szüsz \cite{RS-1992} and obtained the solution of its Gauss-Kuzmin-type problem.
We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of \cite{Magyar_dyadic}, \cite{Magyar_ergodic} and \cite{MSW}. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring's problem to an approximation formula for the fourier transform of the solution set of lattice points on hypersurfaces arising in Waring's problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation form…
In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted by Frantzikinakis (Problem 10, "Some open problems on multiple ergodic averages. Bulletin of the Hellenic Mathematical Society. 60 (2016), 41-90"). Corresponding averages along prime numbers are studied too. These general convergence results imply various variable extensions of classical recurrence, combinatorial and number theoretical results which are prese…
Novikov's problem of semiclassical orbits of quasi-electrons in a normal metal leads to a correspondance between 3-ply periodic functions in R and fractals in R P^2. These fractals are the complement of infinitely many open sets labeled by integer 2-cycles of T^3. Here we present a characterization of the fractal points in terms of the open sets labels.
Invariant measures on the space of measured laminations for subgroups of mapping class group
本文研究曲面映射类群子群在测度叶状空间上的不变Radon测度，通过分类回归叶状空间上的测度，证明发散型子群的唯一遍历性并推广至凸余紧子群，方法几何且适用于多种度量空间。
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.
Let G:=SO(n,1)^\circ and Γbe a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(Γ\ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and for delta>n-2 for n>= 4, we obtain an {\it effective} archimedean counting result for a discrete orbit of Γin a homogeneous space H\G, where H is the trivial group, an affine symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family {B_T} of compact subsets in H\G, there exists η>0 such that #[e]\G\cap B_T=M(B_…
In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of $L^p$-norms and leave its a.s. convergence as an open problem. This problem shares some similarities with a well-known unresolved conjecture on a.s. convergence of double ergodic averages with respect to two commuting transformations.
Ergodicity for stochastic T-monotone parabolic obstacle problems
本文研究了由Sobolev空间上T-单调算子控制、受乘性噪声驱动的障碍问题，证明了其解定义了Markov-Feller半群，并在适当假设下确立了遍历不变测度的存在性与唯一性。
We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose $σ$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $Π$ the set of all Borel probabilities $π$ on $X\times X$, such that, both its $x$ and $y$ marginal are $σ$-invariant probabilities. We are interested in the optimal p…
Quantitative Convergence for Sparse Ergodic Averages in $L^1$
本文提出了一种统一框架，证明了稀疏序列在 $L^1$ 端点处的点wise收敛性，并给出了新的定量收敛速度估计。
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.
Stationary scattering problem (when the distance $r$ tends to infinity) and dynamical scattering problem (when the time $t$ tends to infinity) are considered for the 3D Schrödinger equation. A simple interconnection between the scattering amplitude (stationary case) and scattering operator (dynamical case) is given in the paper. This result is a quantum mechanical analog of the ergodic formulas in the classical mechanics.
The Ultimate Ishii-Pastur Theorem for Whole-Line Ergodic Block Jacobi Operators
研究全直线遍历块雅可比算子的终极石井 - 帕斯特定理，通过证明最大谱测度在特定区域的限制由零容量博雷尔集承载，在标量情形下解决了相关全直线问题。
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