Measure-preserving System Research Articles

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set$R\subset \mathbb{N}$is calledrationalif it is well approximable by finite unions of arithmetic progressions, meaning that for every$\unicode[STIX]{x1D716}>0$there exists a set$B=\bigcup _{i=1}^{r}a_{i}\mathbb{N}+b_{i}$, where$a_{1},\ldots ,a_{r},b_{1},\ldots ,b_{r}\in \mathbb{N}$, such that$$\begin{eqnarray}\overline{d}(R\triangle B):=\limsup _{N\rightarrow \infty }\frac{|(R\triangle B)\cap \{1,\ldots ,N\}|}{N}<\unicode[STIX]{x1D716}.\end{eqnarray}$$Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form$\unicode[STIX]{x1D6F7}_{x}:=\{n\in \mathbb{N}:\boldsymbol{\unicode[STIX]{x1D711}}(n)/n<x\}$, where$x\in [0,1]$and$\boldsymbol{\unicode[STIX]{x1D711}}$is Euler’s totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if$R\subset \mathbb{N}$is a rational set with$\overline{d}(R)>0$, then the following are equivalent:(a)$R$is divisible, i.e. $\overline{d}(R\cap u\mathbb{N})>0$for all$u\in \mathbb{N}$;(b)$R$is an averaging set of polynomial single recurrence;(c)$R$is an averaging set of polynomial multiple recurrence.As an application, we show that if$R\subset \mathbb{N}$is rational and divisible, then for any set$E\subset \mathbb{N}$with$\overline{d}(E)>0$and any polynomials$p_{i}\in \mathbb{Q}[t]$,$i=1,\ldots ,\ell$, which satisfy$p_{i}(\mathbb{Z})\subset \mathbb{Z}$and$p_{i}(0)=0$for all$i\in \{1,\ldots ,\ell \}$, there exists$\unicode[STIX]{x1D6FD}>0$such that the set$$\begin{eqnarray}\{n\in R:\overline{d}(E\cap (E-p_{1}(n))\cap \cdots \cap (E-p_{\ell }(n)))>\unicode[STIX]{x1D6FD}\}\end{eqnarray}$$has positive lower density.Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involverationally almost periodic sequences(sequences whose level-sets are rational). We prove that if${\mathcal{A}}$is a finite alphabet,$\unicode[STIX]{x1D702}\in {\mathcal{A}}^{\mathbb{N}}$is rationally almost periodic,$S$denotes the left-shift on${\mathcal{A}}^{\mathbb{Z}}$and$$\begin{eqnarray}X:=\{y\in {\mathcal{A}}^{\mathbb{Z}}:\text{each word appearing in}~y~\text{appears in}~\unicode[STIX]{x1D702}\},\end{eqnarray}$$then$\unicode[STIX]{x1D702}$is a generic point for an$S$-invariant probability measure$\unicode[STIX]{x1D708}$on$X$such that the measure-preserving system$(X,\unicode[STIX]{x1D708},S)$is ergodic and has rational discrete spectrum.

Ergodicity of the Liouville system implies the Chowla conjecture

Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Analysis 2017:19, 41 pp. The Liouville function $\lambda:\mathbb N\to\{-1,1\}$ takes a product $p_1p_2\dots p_k$ of (not necessarily distinct) primes $p_1,\dots,p_k$ to $(-1)^k$. That is, $\lambda(n)$ is the parity of the number of primes in the prime decomposition of $n$. It is a completely multiplicative function (that is, $\lambda(mn)=\lambda(m)\lambda(n)$ for any two positive integers $m,n$, and like various other multiplicative functions it plays an important role in number theory. In particular, it appears to behave like a random $\pm 1$ sequence in various ways, and if this could be proved rigorously it would have major consequences: the Riemann hypothesis, for instance, is equivalent to the statement that the sums $\lambda(1)+\lambda(2)+\dots+\lambda(n)$ grow more slowly than $n^\alpha$ whenever $\alpha>1/2$. (Note that this would be the expected behaviour for a random sequence.) It is known at least that $\lambda(1)+\dots+\lambda(n)=o(n)$, and in fact this statement is equivalent to the prime number theorem. Of great interest recently has been the study of more detailed statistical properties of the Liouville function, where one is interested not just in individual values but in correlations between nearby values. For example, it is believed that $\lambda(x)$ and $\lambda(x+1)$ should be independent, in the sense that $\lambda(1)\lambda(2)+\dots+\lambda(n)\lambda(n+1)=o(n)$. More generally, the Chowla conjecture asserts that for any set $\{k_1,\dots,k_r\}$ of distinct positive integers we have that $\sum_{m\leq n}\lambda(m+k_1)\dots\lambda(m+k_r)=o(n)$. This conjecture is still wide open, though weakenings of it, which have been proved in the last few years by Matomaki, Radziwill, Tao and Teravainen (in various combinations), turn out to be very useful for applications. For example, Tao, building on work by Matomaki and Radziwill, established a logarithmically averaged version of the conjecture for two-point correlations, and used it in [his solution of the Erdos discrepancy problem](http://discreteanalysisjournal.com/article/609-the-erdos-discrepancy-problem). Also, Tao and Teravainen have proved a logarithmically averaged version for all correlations of an odd number of points, which, slightly surprisingly, turns out to be an easier problem. (One might at first think that this would imply the same for all correlations. However, the statement that $\lambda(m+k_1)\dots\lambda(m+k_r)$ averages zero is not the same as the statement that the values $\lambda(m+k_1),\dots,\lambda(m+k_r)$ are independent, and it does not imply independence even if one assumes the same statement for all subsets of $\{k_1,\dots,k_r\}$ of odd size, or indeed for all subsets of $\mathbb N$ of odd size.) It turns out to be rather fruitful to reformulate questions of this type as questions about dynamical systems. A standard construction in symbolic dynamics is to take an infinite sequence of letters in a finite alphabet and take the closure of all its shifts (in the product topology). The shift acts on this closure, turning it into a dynamical system. Thanks to the Furstenberg correspondence principle, it comes with a natural measure that makes it into a measure-preserving system. The properties of this system relate to various features of the sequence, and methods from ergodic theory and topological dynamics can be used to prove interesting theorems that do not involve dynamics in their statements. The main result of this paper is that the general logarithmically averaged Chowla conjecture (that is, the statement for all correlations and not just correlations between pairs) follows from a dynamical statement that looks significantly weaker. In dynamical terms, the Chowla conjecture states that the dynamical system one obtains from the Liouville sequence has the Bernoulli property, which means that it is isomorphic to the system one gets from all $\pm 1$ sequences. (This implication holds because if every finite string occurs with the right frequency, then shifts of the Liouville sequence land in a basic open neighbourhood with the right frequency, so all the basic open neighbourhoods have the right measure and the system becomes the same as what one would get with the shift acting on the space of all $\pm 1$ sequences with the product measure.) The paper shows that to prove the Bernoulli property, it is enough to establish that the system is ergodic, which means that there are no invariant subsets of measure strictly between 0 and 1. In general, an implication like this is far from true: there are a number of properties that say that a dynamical system is somewhat random: the property of being ergodic is one of the weakest, and the Bernoulli property is the strongest. So the result is telling us something interesting about the Liouville sequence. Several powerful tools from analytic number theory, additive combinatorics, and ergodic theory are used in the proof.

Ruelle's inequality in negative curvature

In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.

Under-recurrence in the Khintchine recurrence theorem

The Khintchine recurrence theorem asserts that in a measure preserving system, for every set A and e > 0, we have μ(A ∩ T−nA) ≥ μ(A)2 − e for infinitely many n ∈ N. We show that there are systems having underrecurrent sets A, in the sense that the inequality μ(A ∩ T−nA) < μ(A)2 holds for every n ∈ N. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V. Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants of these problems where the previous strict inequality is reversed, and deduce that under-recurrence is a much more rare phenomenon than over-recurrence. Finally, we study related problems pertaining to multiple recurrence and derive some interesting combinatorial consequences.

Recurrence, rigidity, and popular differences

We construct a set of integers $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. Here ‘set of rigidity’ means that enumerating $S$ as $(s_{n})_{n\in \mathbb{N}}$ produces a rigidity sequence. This construction generalizes or strengthens results of Katznelson, Saeki (on equidistribution and the Bohr topology), Forrest (on sets of recurrence and strong recurrence), and Fayad and Kanigowski (on rigidity sequences). The construction also provides a density analogue of Julia Wolf’s results on popular differences in finite abelian groups.

Dynamical intricacy and average sample complexity

ABSTRACTWe propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define intricacy and average sample complexity for topological and measure-preserving dynamical systems. We establish basic properties of these quantities, show that their suprema over covers or partitions equal the ordinary entropies, compute them for many shifts of finite type, and indicate natural directions for further research.

A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

We investigate how spectral properties of a measure-preserving system$(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$are reflected in the multiple ergodic averages arising from that system. For certain sequences$a:\mathbb{N}\rightarrow \mathbb{N}$, we provide natural conditions on the spectrum$\unicode[STIX]{x1D70E}(T)$such that, for all$f_{1},\ldots ,f_{k}\in L^{\infty }$,$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{ja(n)}f_{j}=\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{jn}f_{j}\end{eqnarray}$$in$L^{2}$-norm. In particular, our results apply to infinite arithmetic progressions,$a(n)=qn+r$, Beatty sequences,$a(n)=\lfloor \unicode[STIX]{x1D703}n+\unicode[STIX]{x1D6FE}\rfloor$, the sequence of squarefree numbers,$a(n)=q_{n}$, and the sequence of prime numbers,$a(n)=p_{n}$. We also obtain a new refinement of Szemerédi’s theorem via Furstenberg’s correspondence principle.

Almost local stability in discrete delayed chaotic systems

This work studies dynamic of delayed discrete chaotic systems with bounded and unbounded delays. The time lags appear in additive which is coupled with a smooth function and nonadditive forms. It has been shown that, in both additive and nonadditive cases, the primal (non-delayed) system is neutral to the bounded delay to possess an attractive fixed point. Nevertheless, if a nonadditive and unbounded delay is supposed to affect a chaotic and measure preserving system locally, then the delay function might be sensitive to initial states. A local stabilization to the dynamics of Logistic and Gaussian maps are made and creation of attractive fixed points is illustrated.

Quantitative Pesin theory for Anosov diffeomorphisms and flows

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure-preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows and consider counterexamples illustrating the necessity of suitable conditions on the cocycle.

Poincaré recurrence theorem in impulsive systems

In this article, we generalize the Poincaré recurrence theorem to impulsive dynamical systems in $\mathbb R^n$. For a measure preserving system, we present some sufficient conditions to establish an impulsive system that is also measure preserving. Then, two recurrence theorems are proved. Finally, we use two examples to illustrate our results.

Constructive symbolic presentations of rank one measure-preserving systems

Given a rank one measure-preserving system defined by cutting and stacking with spacers, we produce a rank one binary sequence such that its orbit closure under the shift transformation, with its unique nonatomic invariant probability, is isomorphic to the given system. In particular, the classical dyadic odometer is presented in terms of a recursive sequence of blocks on the two-symbol alphabet {0, 1}. The construction is accomplished using a definition of rank one in the setting of adic, or Bratteli-Vershik, systems.

Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

On the limits of probabilistic forecasting in nonlinear times series analysis.

The ignorance score measures the quality of probabilistic forecasting. In this paper, we study its basic properties in the perfect model scenario, i.e., under the assumption that the system producing the data is perfectly known. Two further qualifications are added to this general setting. First, the system is a discrete-time, measure-preserving dynamical system. Moreover, randomness results from the quantization of the state space (i.e., from the finite precision of the observations), rather than being introduced via observational noise. In this "non-linear" perfect model scenario we derive, in particular, the admissible domain of the ignorance score and relate it with the ignorance score in imperfect models.

Escape Rates for the Farey Map with Approximated Holes

We study the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. Due to the ergodic properties of the map, the standard theoretical approaches to this problem cannot be applied. It has been recently shown in [Knight &amp; Munday, 2016] how to apply the standard analytical methods to a piecewise-linear version of the Farey map with holes depending on the associated partition, but their results cannot be obtained in the general case we consider here. To overcome these difficulties we propose here to study approximations of the hole by means of real analytic functions. We introduce a particular family of approximations and study numerically the behavior of the escape rate for approximated holes with vanishing measure. The results suggest that the scaling of the escape rate depends on the “shape” of the approximation, and we show that this is a typical feature of systems with an indifferent fixed point, not an artifact of the particular family we consider.

Periodic points and the measure of maximal entropy of an expanding Thurston map

In this paper, we show that each expanding Thurston map f : S 2 → S 2 f\colon S^2\!\rightarrow S^2 has 1 + deg ⁡ f 1+\deg f fixed points, counted with appropriate weight, where deg ⁡ f \deg f denotes the topological degree of the map f f . We then prove the equidistribution of preimages and of (pre)periodic points with respect to the unique measure of maximal entropy μ f \mu _f for f f . We also show that ( S 2 , f , μ f ) (S^2,f,\mu _f) is a factor of the left shift on the set of one-sided infinite sequences with its measure of maximal entropy, in the category of measure-preserving dynamical systems. Finally, we prove that μ f \mu _f is almost surely the weak ∗ ^* limit of atomic probability measures supported on a random backward orbit of an arbitrary point.

Uniformly Distributed Systems of Elements on Metabelian Lie Rings

In this article, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main result of the article, it was shown that on the variety of metabelian Lie algebras a system of elements is primitive iff it is uniformly distributed.

On Comparison of Dynamics of Dissipative and Finite-Time Systems Using Koopman Operator Methods

Abstract: Comparison of different dynamical systems or a dynamical system with data is one of the core issues in both dynamical systems and control theory. The fruitful approaches are particularly hard to come by for systems and data that show nonlinear behavior and non-Gaussian noise characteristics. We describe a theory that utilizes spectral theory of linear operators (composition, or Koopman) to provide the methodology, that was originally formulated for measure-preserving systems. Here we extend it to capture dissipative and finite-time dynamics. The approach combines a version of ergodic partition theory with Hardy space theory in observable space (rather than in time).

A monotone Sinai theorem

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.

A good universal weight for nonconventional ergodic averages in norm

We will show that the sequence appearing in the double recurrence theorem is a good universal weight for the Furstenberg averages. That is, given a system$(X,{\mathcal{F}},\unicode[STIX]{x1D707},T)$and bounded functions$f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$, there exists a set of full-measure$X_{f_{1},f_{2}}$in$X$that is independent of integers$a$and$b$and a positive integer$k$such that, for all$x\in X_{f_{1},f_{2}}$, for every other measure-preserving system$(Y,{\mathcal{G}},\unicode[STIX]{x1D708},S)$and for each bounded and measurable function$g_{1},\ldots ,g_{k}\in L^{\infty }(\unicode[STIX]{x1D708})$, the averages$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)g_{1}\circ S^{n}g_{2}\circ S^{2n}\cdots g_{k}\circ S^{kn}\end{eqnarray}$$converge in$L^{2}(\unicode[STIX]{x1D708})$.

FAILURE OF THE POINTWISE AND MAXIMAL ERGODIC THEOREMS FOR THE FREE GROUP

Let $F_{2}$ denote the free group on two generators $a$ and $b$. For any measure-preserving system $(X,{\mathcal{X}},{\it\mu},(T_{g})_{g\in F_{2}})$ on a finite measure space $X=(X,{\mathcal{X}},{\it\mu})$, any $f\in L^{1}(X)$, and any $n\geqslant 1$, define the averaging operators $$\begin{eqnarray}\displaystyle {\mathcal{A}}_{n}f(x):=\frac{1}{4\times 3^{n-1}}\mathop{\sum }_{g\in F_{2}:|g|=n}f(T_{g}^{-1}x), &amp; &amp; \displaystyle \nonumber\end{eqnarray}$$ where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f\in L^{1}(X)$ such that the sequence ${\mathcal{A}}_{n}f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maximal ergodic theorems do not hold in $L^{1}$ for actions of $F_{2}$. This is despite the results of Nevo–Stein and Bufetov, who establish pointwise and maximal ergodic theorems in $L^{p}$ for $p&gt;1$ and for $L\log L$ respectively, as well as an estimate of Naor and the author establishing a weak-type $(1,1)$ maximal inequality for the action on $\ell ^{1}(F_{2})$. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.