We investigate limit theorems for Birkhoff sums of locally Hölder functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L^2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem.
We show that there exists a universal gap in the failure of the ergodic theorem for symmetric Birkhoff sums in infinite ergodic theory. In addition, an application of this result to a question of fluctuations of the Birkhoff integrals of horocyclic flows on geometrically finite surfaces is given.
We investigate the growth rate of the Birkhoff sums $S_{n,α} f(x)=\sum_{k=0}^{n-1} f(x+kα)$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb T$ and $(α,x)$ is a "typical" element of $\mathbb T^2$. The answer depends on the meaning given to the word "typical". Part of the work will be done in a more general context.
We consider implications of dynamical Borel-Cantelli lemmas for rates of growth of Birkhoff sums of non-integrable observables $φ(x) = d(x,p)^{-k}$, $k>0$, on ergodic dynamical systems $(T,X,μ)$ where $μ(X) = 1$. Some general results are given as well as some more concrete examples involving non-uniformly expanding maps, intermittent type maps as well as uniformly hyperbolic systems.
We prove strong laws of large numbers under intermediate trimming for Birkhoff sums over subshifts of finite type. This gives another application of a previous trimming result only proven for interval maps. In case of Markov measures we give a further example of St.\ Petersburg type distribution functions. To prove these statements we introduce the space of quasi-Hölder continuous functions for subshifts of finite type.
Often topological classes of one-dimensional dynamical systems are finite codimension smooth manifolds. We describe a method to prove this sort of statement that we believe can be applied in many settings. In this work we will implement it for piecewise expanding maps. The most important step will be the identification of infinitesimal deformations with primitives of Birkhoff sums (up to addition of a Lipschitz function), that allows us to use the ergodic properties of piecewise expanding maps to study the regularity of infinitesimal deformations.
We consider dynamical systems on a finite measure space fulfilling a spectral gap property and Birkhoff sums of a non-negative, non-integrable observable. For such systems we generalize strong laws of large numbers for intermediately trimmed sums only known for independent random variables. The results split up in trimming statements for general distribution functions and for regularly varying tail distributions. In both cases the trimming rate can be chosen in the same or almost the same way as in the i.i.d. case. As an example we show that piecewise expanding interval maps fulfill the necess…
We study Birkhoff sums S(k,a) = g(a)+g(2a)+...+g(ka) at the golden mean rotation number a with periodic continued fraction approximations p(n)/q(n), where g(x) = log(2-2 cos(2 pi x). The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena. We relate the boundedness of log- averaged Birkhoff sums S(k,a)/log(k) and the convergence of S(q(n),a) with the existence of an experimentally established limit function f(x) = lim S([x q(n)])(p(n+1)/q(n+1))-S([x q(n)])(p(n)/q(n)) for n to infinity on the interval [0,1]. The function f satisfies a functional equa…
This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions …
The increasing rate of the Birkhoff sums in the infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate are obtained.
We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.
We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdős, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in $(0,1)$.
Let $((0,1], T)$ be the doubling map in the unit interval and $φ$ be the Saint-Petersburg potential, defined by $φ(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=φ(x)+\cdots+φ(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim\_{n\to\infty}S\_n(x)/n=α\} \ (α>0)$, as well as that of the set $\{x: \lim\_{n\to\inft…
On a measure theoretical dynamical system with spectral gap property we consider non-integrable observables with regularly varying tails and fulfilling a mild mixing condition. We show that the normed trimmed sum process of these observables then converges in mean. This result is new also for the special case of i.i.d. random variables and contrasts the general case where mean convergence might fail even though a strong law of large numbers holds. To illuminate the required mixing condition we give an explicit example of a dynamical system fulfilling a spectral gap property and an observable w…
A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long time behavior of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew product dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounov exponents. The sequence, taking the values $\pm 1$, is pairwise independent (but not independent) ergodic sequence with infinite range dependence. The model corresponds to the motion of a particle on an infinite cylinder, hopping backward and forward along its axis, with a transversal acceleration p…
We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(α,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point, $α\in [0,1)$ is the rotation number and $(α, x)$ are uniformly distributed. We prove that $S_N/N$ has a joint limiting distribution in $(α,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.
In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $ϕ$ is an Hölder continuous potential we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb R$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we de…
We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. U…
Birkhoff Measures, Birkhoff Sums, and Discrepancies
SUNY Old Westbury（纽约州立大学奥尔德韦斯伯里分校）；FM Tangerman LLC（FM Tangerman有限责任公司）；Portland State University（波特兰州立大学）；Inst. of Analysis and Number Theory, Graz UT（格拉茨技术大学分析与数论研究所）
We consider the deviation of Birkhoff sums along fixed orbits of substitution dynamical systems. We show distributional convergence for the Birkhoff sums of eigenfunctions of the substitution matrix. For noncoboundary eigenfunctions with eigenvalue of modulus 1, we obtain a central limit theorem. For other eigenfunctions, we show convergence to distributions supported on Cantor sets. We also give a new criterion for such an eigenfunction to be a coboundary, as well as a new characterization of substitution dynamical systems with bounded discrepancy
We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than $\frac{1}{2}$ are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property is well controlled almost everywhere. We prove that for any $q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.
We establish quantitative results for the statistical be\-ha\-vi\-our of \emph{infinite systems}. We consider two kinds of infinite system: i) a conservative dynamical system $(f,X,μ)$ preserving a $σ$-finite measure $μ$ such that $μ(X)=\infty$; ii) the case where $μ$ is a probability measure but we consider the statistical behaviour of an observable $ϕ\colon X\to[0,\infty)$ which is non-integrable: $\int ϕ\, dμ=\infty$. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly …
In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form $x+x^{1+α}$, for $α\in (0,1)$. In particular, for $α>1/2$, we show that the Birkhoff sums of a Hölder observable $f$ converge to a normal law or a stable law, depending on whether $f(0)=0$ or $f(0)\not=0$. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in noncommutative Banach algebras.
We consider a conservative ergodic measure-preserving transformation $T$ of a $σ$-finite measure space $(X,\mathcal{B},μ)$ with $μ(X)=\infty$. Given an observable $f:X\to \mathbb{R}$ we study the almost sure asymptotic behaviour of the Birkhoff sums $S_Nf(x) := \sum_{j=1}^N\, (f\circ T^{j-1})(x)$. In infinite ergodic theory it is well known that the asymptotic behaviour of $S_Nf(x)$ strongly depends on the point $x\in X$, and if $f\in L^1(X,μ)$, then there exists no real valued sequence $(b(N))$ such that $\lim_{N\to\infty} S_Nf(x)/b(N)=1$ almost surely. In this paper we show that for dynamica…
Livšic theorem asserts that, for Anosov diffeomorphisms/flows, a Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic orbits are equal to zero. The transfer function is then Lipschitz. We prove a positive Livšic theorem which asserts that a Lipschitz observable is bounded from below by a coboundary if and only if all its Birkhoff sums on periodic orbits are non negative. The new result is that the coboundary can be chosen Lipschitz. The map is only assumed to be $C^1$ and hyperbolic, but not necessarily bijective nor transitive. We actually prove our main result in t…
We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius $\eps$ have a measure smaller than $\eps^δ$ and half of them have a measure larger than $\eps^δ$, where $δ$ is the Hausdorff dimension of the measure. We first show that the problem is equivalent to the study of t…
We obtain optimal moment bounds for Birkhoff sums, and optimal concentration inequalities, for a large class of slowly mixing dynamical systems, including those that admit anomalous diffusion in the form of a stable law or a central limit theorem with nonstandard scaling $(n\log n)^{1/2}$.
For a class of locally (but not necessarily uniformly) Lipschitz continuous $d$-dimensional observables over a Gibbs-Markov system, we show that convergence of (suitably normalized and centered) ergodic sums to a non-Gaussian stable vector is equivalent to the distribution belonging to the classical domain of attraction, and that it implies a weak invariance principle in the (strong) Skorohod $\mathcal{J}_{1}$-topology on $\mathcal{D}([0,\infty),\mathbb{R}^{d})$. The argument uses the classical approach via finite-dimensional marginals and $\mathcal{J}_{1}$-tightness. As applications, we recor…
Bromberg and Ulcigrai constructed piecewise smooth functions f on the torus such that the set of angles alpha for which the Birkhoff sums of f with respect to the irrational translation by alpha satisfies a temporal distributional limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show that the Lebesgue measure of this set of angles is equal to zero.
In the case of smooth non-invertible maps which are hyperbolic on folded basic sets $Λ$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary Hölder potentials, with the help of weighted sums of atomic measures on preimage sets of high order. Our endomorphism may have also stable directions on $Λ$, thus it is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (they depend on the whole past and may intersect each other both inside and outside $Λ$), and moreover the number of preimages r…
We consider normalizing sequences that can give rise to nondegenerate limittheorems for Birkhoff sums under the iteration of a conservative map. Mostclassical limit theorems involve normalizing sequences that are polynomial,possibly with an additional slowly varying factor. We show that, ingeneral, there can be no nondegenerate limit theorem with a normalizingsequence that grows exponentially, but that there are examples where itgrows like a stretched exponential, with an exponent arbitrarily close to 1.
The work [8] established memory loss in the time-dependent (non-random) case of uniformly expanding maps of the interval. Here we find conditions under which we have convergence to the normal distribution of the appropriately scaled Birkhoff-like partial sums of appropriate test functions. A substantial part of the problem is to ensure that the variances of the partial sums tend to infinity (cf. the zero-cohomology condition in the autonomous case). In fact, the present paper is the first one where non-random, i. e. specific examples are also found, which are not small perturbations of a given…
Eagleson's Theorem asserts that, given a probability-preserving map, ifrenormalized Birkhoff sums of a function converge in distribution, thenthey also converge with respect to any probability measure which isabsolutely continuous with respect to the invariant one. We prove a versionof this result for almost sure limit theorems, extending results ofKorepanov. We also prove a version of this result, in mixing systems, whenone imposes a conditioning both at time 0 and at time n.
A dynamical array consists of a family of functions $\{f_{n,i}: 1\le i\le k(n), n\ge 1\}$ and a family of initial times $\{τ_{n,i}: 1\le i\le k(n), n\ge 1\}$. For a dynamical system $(X,T)$ we identify distributional limits for sums of the form $$ S_n= \frac 1{s_n}\sum_{i=1}^{k(n)} [f_{n,i}\circ T^{τ_{n,i}}-a_{n,i}]\qquad n\ge 1$$ for suitable (non-random) constants $s_n>0$ and $a_{n,i}\in \mathbb R$, where the functions $f_{n,i}$ are locally Lipschitz continuous. Although our results hold for more general dynamics, we restrict to Gibbs-Markov dynamical systems for convenience. In particular, …
Let $A$ be a finite set and $ϕ:A^Z\to R$ be a locally constant potential. For each $β>0$ ("inverse temperature"), there is a unique Gibbs measure $μ_{βϕ}$. We prove that, as $β\to+\infty$, the family $(μ_{βϕ})_{β>0}$ converges (in weak-$^*$ topology) to a measure we characterize. It is concentrated on a certain subshift of finite type which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron-Frobenius Theorem for matrices á la Birkhoff. The crucial idea we bring is a "renormalization" procedure which explains conve…
We prove that the Birkhoff sums for ``almost every'' relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a $\sqrt{n\log n}$ normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.
We prove a central limit theorem for Birkhoff sums of the Rosen continued fraction algorithm. A Lasota-Yorke bound is obtained for general one-dimensional continued fractions with the bounded variation space, which implies quasi-compactness of the transfer operator. The main result is a direct proof of the existence of a spectral gap, assuming a certain behavior of the transformation when iterated. This condition is explicitly proved for the Rosen system. We conclude via well-known results of A. Broise that the central limit theorem holds.
This paper is devoted to the study of the stochastic properties of dynamical systems preserving an infinite measure. More precisely we prove central limit theorems for Birkhoff sums of observables of $\mathbb{Z}^2$-extensions of dynamical systems (satisfying some nice spectral properties). The motivation of our paper is the $\mathbb{Z}^2$-periodic Lorentz process for which we establish a functional central limit theorem for Hölder continuous observables (in discrete time as well as in continuous time).
We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-like partial sums, with estimates on the rate of convergence. For maps chosen from a certain parameter range, but without additional assumptions on how the maps vary with time, we obtain a self-normalized CLT provided that the variances of the partial sums grow sufficiently fast. When the maps are chosen randomly according to a shift-invariant probability measure, we identify conditions under which the …
We study higher order expansions both in the Berry-Esséen estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting.
We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in $(0,1]$ ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We show that central values of th…
Consider a nonuniformly hyperbolic map $ T $ modelled by a Young tower with tails of the form $ O(n^{-β}) $, $ β>2 $. We prove optimal moment bounds for Birkhoff sums $ \sum_{i=0}^{n-1}v\circ T^i $ and iterated sums $ \sum_{0\le i<j<n}v\circ T^i\, w\circ T^j $, where $ v,w:M\to \Bbb{R}$ are (dynamically) Hölder observables. Previously iterated moment bounds were only known for $ β>5$. Our method of proof is as follows; (i) prove that $ T $ satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimat…
We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence $b(N)$ of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence $b(N)$, the conditions imply that the error in the approximation decays either at the rate $O(N^{-1/2})$ or the rate $O(N^{-1/2} \log N)$, under the additional …
