Measure-preserving System Research Articles

Under- and over-independence in measure preserving systems

We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. Here is a sample of results obtained in this paper: $\cdot$ (Existence of density-1 UI and OI set) Let $(X,\mathcal{B},\mu,T)$ be an invertible probability measure preserving weakly mixing system. Then for any $d\in\mathbb{N}$, any non-constant integer-valued polynomials $p_{1},p_{2},\dots,p_{d}$ such that $p_{i}-p_{j}$ are also non-constant for all $i\neq j$, (i) there is $A\in\mathcal{B}$ such that the set $$\{n\in\mathbb{N}\colon\mu(A\cap T^{p_{1}(n)}A\cap\dots\cap T^{p_{d}(n)}A)<\mu(A)^{d+1}\}$$ is of density 1. (ii) there is $A\in\mathcal{B}$ such that the set $$\{n\in\mathbb{N}\colon\mu(A\cap T^{p_{1}(n)}A\cap\dots\cap T^{p_{d}(n)}A)>\mu(A)^{d+1}\}$$ is of density 1. $\cdot$ (Existence of Ces\`aro OI set) Let $(X,\mathcal{B},\mu,T)$ be a free, invertible, ergodic probability measure preserving system and $M\in\mathbb{N}$. %Suppose that $X$ contains an ergodic component which is aperiodic. Then there is $A\in\mathcal{B}$ such that $$\frac{1}{N}\sum_{n=M}^{N+M-1}\mu(A\cap T^{n}A)>\mu(A)^{2}$$ for all $N\in\mathbb{N}$. $\cdot$ (Nonexistence of Ces\`aro UI set) Let $(X,\mathcal{B},\mu,T)$ be an invertible probability measure preserving system. For any measurable set $A$ satisfying $\mu(A) \in (0,1)$, there exist infinitely many $N \in \mathbb{N}$ such that $$\frac{1}{N} \sum_{n=0}^{N-1} \mu ( A \cap T^{n}A) > \mu(A)^2.$$

Spectrum of the Koopman Operator, Spectral Expansions in Functional Spaces, and State-Space Geometry

We examine spectral operator-theoretic properties of linear and nonlinear dynamical systems with globally stable attractors. Using the Kato decomposition, we develop a spectral expansion for general linear autonomous dynamical systems with analytic observables and define the notion of generalized eigenfunctions of the associated Koopman operator. We interpret stable, unstable and center subspaces in terms of zero-level sets of generalized eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions and the new notion of open eigenfunctions—defined on subsets of state space—to extend these results to nonlinear dynamical systems with an equilibrium. We provide a characterization of (global) center manifolds, center-stable, and center-unstable manifolds in terms of joint zero-level sets of families of Koopman operator eigenfunctions associated with the nonlinear system. After defining a new class of Hilbert spaces, that capture the on- and off-attractor properties of dissipative dynamics, and introducing the concept of modulated Fock spaces, we develop spectral expansions for a class of dynamical systems possessing globally stable limit cycles and limit tori, with observables that are square-integrable in on-attractor variables and analytic in off-attractor variables. We discuss definitions of stable, unstable, and global center manifolds in such nonlinear systems with (quasi)-periodic attractors in terms of zero-level sets of Koopman operator eigenfunctions. We define the notion of isostables for a general class of nonlinear systems. In contrast with the systems that have discrete Koopman operator spectrum, we provide a simple example of a measure-preserving system that is not chaotic but has continuous spectrum, and discuss experimental observations of spectrum on such systems. We also provide a brief characterization of the data types corresponding to the obtained theoretical results and define the coherent principal dimension for a class of datasets based on the lattice-type principal spectrum of the associated Koopman operator.

Weak ergodic averages over dilated measures

Let$m\in \mathbb{N}$and$\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$be a measure-preserving system with an$\mathbb{R}^{m}$-action. We say that a Borel measure$\unicode[STIX]{x1D708}$on$\mathbb{R}^{m}$is weakly equidistributed for$\mathbf{X}$if there exists$A\subseteq \mathbb{R}$of density 1 such that, for all$f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$for$\unicode[STIX]{x1D707}$-almost every$x\in X$. Let$W(\mathbf{X})$denote the collection of all$\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$such that the$\mathbb{R}$-action$(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$is not ergodic. Under the assumption of the pointwise convergence of the double Birkhoff ergodic average, we show that a Borel measure$\unicode[STIX]{x1D708}$on$\mathbb{R}^{m}$is weakly equidistributed for an ergodic system$\mathbf{X}$if and only if$\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$for every$\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$. Under the same assumption, we also show that$\unicode[STIX]{x1D708}$is weakly equidistributed for all ergodic measure-preserving systems with$\mathbb{R}^{m}$-actions if and only if$\unicode[STIX]{x1D708}(\ell )=0$for all hyperplanes$\ell$of $\mathbb{R}^{m}$. Unlike many equidistribution results in literature whose proofs use methods from harmonic analysis, our results adopt a purely ergodic-theoretic approach.

Optimal lower bounds for multiple recurrence

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}&gt;0$. We study the largeness of sets of the form $$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)&gt;\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$ for various families $\{f_{1},\ldots ,f_{k}\}$ of sequences $f_{i}:\mathbb{N}\rightarrow \mathbb{N}$. For $k\leq 3$ and $f_{i}(n)=if(n)$, we show that $S$ has positive density if $f(n)=q(p_{n})$, where $q\in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1)=0$ and $p_{n}$ denotes the $n$th prime; or when $f$ is a certain Hardy field sequence. If $T^{q}$ is ergodic for some $q\in \mathbb{N}$, then, for all $r\in \mathbb{Z}$, $S$ is syndetic if $f(n)=qn+r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and, for $k=3$, we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.

Quantum mechanics and data assimilation.

A framework for data assimilation combining aspects of operator-theoretic ergodic theory and quantum mechanics is developed. This framework adapts the Dirac-von Neumann formalism of quantum dynamics and measurement to perform sequential data assimilation (filtering) of a partially observed, measure-preserving dynamical system, using the Koopman operator on the L^{2} space associated with the invariant measure as an analog of the Heisenberg evolution operator in quantum mechanics. In addition, the state of the data assimilation system is represented by a trace-class operator analogous to the quantum mechanical density operator, and the assimilated observables by self-adjoint multiplication operators. An averaging approach is also introduced, rendering the spectrum of the assimilated observables discrete and thus amenable to numerical approximation. We present a data-driven formulation of the quantum mechanical data assimilation approach, utilizing kernel methods from machine learning and delay-coordinate maps of dynamical systems to represent the evolution and measurement operators via matrices in a data-driven basis. The data-driven formulation is structurally similar to its infinite-dimensional counterpart and shown to converge in a limit of large data under mild assumptions. Applications to periodic oscillators and the Lorenz 63 system demonstrate that the framework is able to naturally handle highly non-Gaussian statistics, complex state space geometries, and chaotic dynamics.

Measure-theoretic mean equicontinuity and bounded complexity

Let (X,B,μ,T) be a measure preserving system. We say that a function f∈L2(X,μ) is μ-mean equicontinuous if for any ϵ>0 there is k∈N and measurable sets A1,A2,⋯,Ak with μ(⋃i=1kAi)>1−ϵ such that whenever x,y∈Ai for some 1≤i≤k, one haslimsupn→∞1n∑j=0n−1|f(Tjx)−f(Tjy)|<ϵ. Measure-theoretic complexity with respect to f is also introduced. It is shown that f is an almost periodic function if and only if f is μ-mean equicontinuous if and only if μ has bounded complexity with respect to f.Ferenczi studied measure-theoretic complexity using α-names of a partition and the Hamming distance. He proved that if a measure preserving system is ergodic, then the complexity function is bounded if and only if the system has discrete spectrum. We show that this result holds without the assumption of ergodicity.

Finitary isomorphisms of some infinite entropy Bernoulli flows

A consequence of Ornstein theory is that the infinite entropy flows associated with Poisson processes and continuous-time irreducible Markov chains on a finite number of states are isomorphic as measure-preserving systems. We give an elementary construction of such an isomorphism that has an additional finitariness property, subject to the additional conditions that the Markov chain has a uniform holding rate and a mixing skeleton.

Furstenberg Systems of Bounded Multiplicative Functions and Applications

Abstract We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We deduce that aperiodic multiplicative functions with values plus or minus one have super-linear block growth. Another consequence of our structural result is that products of shifts of arbitrary multiplicative functions with values on the unit disc do not correlate with any totally ergodic deterministic sequence of zero mean. Our methodology is based primarily on techniques developed in a previous article of the authors where analogous results were proved for the Möbius and the Liouville function. A new ingredient needed is a result obtained recently by Tao and Teräväinen related to the odd order cases of the Chowla conjecture.

On convergence of oscillatory ergodic Hilbert transforms

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let $p(t)$ be a Hardy field function which grows super-linearly and stays sufficiently far from polynomials. We show that for each measure-preserving system, $(X,\Sigma,\mu,\tau)$, with $\tau$ a measure-preserving $\mathbb{Z}$-action, the modulated one-sided ergodic Hilbert transform \[ \sum_{n=1}^\infty \frac{e^{2\pi i p(n)}}{n} \tau^n f(x) \] converges $\mu$-a.e. for each $f \in L^r(X), 1 \leq r < \infty$. This affirmatively answers a question of J. Rosenblatt. In the second part of the paper, we establish almost sure sparse bounds for random one-sided ergodic Hilbert transforms, \[ \sum_{n=1}^\infty \frac{X_n}{n} \tau^n f(x), \] where $\{ X_n \}$ are uniformly bounded, independent, and mean-zero random variables.

On the isomorphism problem for non-minimal transformations with discrete spectrum

The article addresses the isomorphism problem for non-minimal topological dynamical systems with discrete spectrum, giving a solution under appropriate topological constraints. Moreover, it is shown that trivial systems, group rotations and their products, up to factors, make up all systems with discrete spectrum. These results are then translated into corresponding results for non-ergodic measure-preserving systems with discrete spectrum.

Stationary coalescing walks on the lattice

We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In $2d$, we classify the collective behavior of these walks under mild assumptions: they either coalesce almost surely or form bi-infinite trajectories. Bi-infinite trajectories form measure-preserving dynamical systems, have a common asymptotic direction in $2d$, and possess other nice properties. We use our theory to classify the behavior of compatible families of semi-infinite geodesics in stationary first- and last-passage percolation. We also partially answer a question raised by C. Hoffman about the limiting empirical measure of weights seen by geodesics. We construct several examples: our main example is a standard first-passage percolation model where geodesics coalesce almost surely, but have no asymptotic direction or average weight.

Zero-dimensional isomorphic dynamical models

By an assignment we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems obeying some natural restrictions. We prove that if $\unicode[STIX]{x1D6F7}$ is an aperiodic assignment on a Choquet simplex $K$ such that the set of extreme points $\mathsf{ex}K$ is a countable union $\bigcup _{n}E_{n}$, where each set $E_{n}$ is compact, zero-dimensional and the restriction of $\unicode[STIX]{x1D6F7}$ to the Bauer simplex $K_{n}$ spanned by $E_{n}$ can be ‘embedded’ in some topological dynamical system, then $\unicode[STIX]{x1D6F7}$ can be ‘realized’ in a zero-dimensional system.

Pointwise multiple averages for sublinear functions

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

A Brief Review of Generalized Entropies.

Entropy appears in many contexts (thermodynamics, statistical mechanics, information theory, measure-preserving dynamical systems, topological dynamics, etc.) as a measure of different properties (energy that cannot produce work, disorder, uncertainty, randomness, complexity, etc.). In this review, we focus on the so-called generalized entropies, which from a mathematical point of view are nonnegative functions defined on probability distributions that satisfy the first three Shannon–Khinchin axioms: continuity, maximality and expansibility. While these three axioms are expected to be satisfied by all macroscopic physical systems, the fourth axiom (separability or strong additivity) is in general violated by non-ergodic systems with long range forces, this having been the main reason for exploring weaker axiomatic settings. Currently, non-additive generalized entropies are being used also to study new phenomena in complex dynamics (multifractality), quantum systems (entanglement), soft sciences, and more. Besides going through the axiomatic framework, we review the characterization of generalized entropies via two scaling exponents introduced by Hanel and Thurner. In turn, the first of these exponents is related to the diffusion scaling exponent of diffusion processes, as we also discuss. Applications are addressed as the description of the main generalized entropies advances.

A double return times theorem

We prove that for any bounded functions $f_1, f_2$ on a measure-preserving dynamical system $(X,T)$ and any distinct integers $a_1, a_2$, for almost every $x$ the sequence $$ f_1(T^{a_1 n}x) f_2(T^{a_2 n}x) $$ is a good weight for the pointwise ergodic theorem.

Multiple Ergodicity For Reversible Markov Chains

Inspired by the multiple recurrence and multiple ergodic theorems for measure preserving systems, we discuss an analogous question for measure preserving semigroups. In this note, we deal with the symmetric semigroups associated to reversible Markov chains.

Möbius disjointness for topological models of ergodic measure-preserving systems with quasi-discrete spectrum

By using a decomposition result for ergodic measure-preserving system with quasi-discrete spectrum, we prove that a generic point of an ergodic quasi-discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that Sarnak's Möbius Disjointness Conjecture holds for any topological model of an ergodic measure-preserving system with quasi-discrete spectrum.

Möbius disjointness for models of an ergodic system and beyond

Given a topological dynamical system (X, T) and an arithmetic function u: ℕ → ℂ, we study the strong MOMO property (relatively to u) which is a strong version of u-disjointness with all observable sequences in (X, T). It is proved that, given an ergodic measure-preserving system (Z, $$\mathcal{D}$$ , к, R),the strong MOMO propertly (relately to u) of a uniquely ergodic midel (X, T)of R yields all other uniquely ergodic midel of R to be u-disjiont. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue—Viorse and Rudin—Shapiro substitutions), systems determined by Kakutani sequences are Mobius (and Liouville) disjoint. The validity of Sarnak5s conjecture implies the strong MOMO property relatively to μ in all zero entropy systems; in particular, it makes μ-disjointness uniform. The absence of the strong MOMO property in positive entropy systems is discussed and it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.

Entropy dimension of measure preserving systems

The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight to analyze the entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate the joinings among entropy zero systems and prove the disjointness property among entropy zero systems using the dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.

The logarithmic Sarnak conjecture for ergodic weights

The Mobius disjointness conjecture of Sarnak states that the Mobius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the Mobius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.