Quantitative uniform exponential acceleration of averages along decaying waves

As one of the foundational theories in dynamical systems, ergodic theory first originated from Birkhoff and von Neumann. This seminal work has given rise to an array of diverse formulations and has been extensively applied across various domains. It is a well-established fact, as noted by Krengel [16], that the convergence rates of Birkhoff averages in ergodic theory are generally slow, and can even be arbitrarily slow for certain counterexamples. Similar assertions were reaffirmed by Ryzhikov [36], while recent results on the convergence rate within the Birkhoff ergodic theorem can be found in Podvigin’s comprehensive survey [24]. Furthermore, Yoccoz’s counterexamples [39, 40], constructed via rotations that are extremely Liouvillean–nearly rational–on the finite-dimensional torus, deserve mentioning. Such slow convergence is ubiquitous in ergodic theory and is intrinsically inescapable, and it would be at most $\mathcal{O}(N^{-1})$ in non-trivial cases with time length $N$, i.e., the observables are non-constant, as detailed by Kachurovskiĭ [13]. More frustratingly, the pursuit of high-precision numerical results may demand computational durations on the order of billions of years, a point elaborated by Das and Yorke in [5].

Given the acknowledged slow convergence in ergodic theory, the weighting method is crucial for accelerating computations in both mathematics and mechanics. There is a growing interest in identifying suitable weighting functions to enhance the convergence rates of ergodic averages. To investigate quasi-periodic perturbations of quasi-periodic flows, Laskar employed a weighting function $\sin^{2}(\pi x)$ to accelerate computational processes, as detailed in [17, 18, 19]. Furthermore, Laskar claimed that a specific exponential weighting function possesses superior asymptotic characteristics, although he did not implement it or provide evidence of its convergence properties, as noted in [19]. It is worth highlighting that the resulting convergence rate can be demonstrated to exceed that of any arbitrary polynomial, which we shall detail later. To be more precise, Laskar employed the following weighting function to investigate ergodicity within dynamical systems:

and the corresponding weighted Birkhoff average of an sufficiently smooth observable $f$ evaluated along a trajectory of length $N$ reads

provided ${A_{N}}:=\sum\nolimits_{n=0}^{N-1}{w\left({n/N}\right)}$, and $T$ is a quasi-periodic mapping on the $d$-torus $\mathbb{T}^{d}$ with $d\in\mathbb{N}^{+}$, $x\in\mathbb{T}^{d}$ is an initial point. It is evident that $w\in C_{0}^{\infty}\left({\left[{0,1}\right]}\right)$ and the integral $\int_{0}^{1}{w\left(x\right)dx}=1$. From a statistical standpoint, this approach effectively reduces the impact of the initial and terminal data, thereby highlighting those in the central range. This emphasis aligns with the concept of averaging, consequently leading to an intuitively rapid uniform convergence to the spatial average $\int_{{\mathbb{T}^{d}}}{f\left(\theta\right)d\theta}$. However, there are more profound reasons behind this phenomenon, which have been illustrated by the authors in [34].

Back to the main topic, following Laskar’s significant numerical discovery, Das and Yorke [5] provided rigorous proofs for quasi-periodic systems with Diophantine rotations. To be more precise, they demonstrated that under $C^{\infty}$ observables, the weighted Birkhoff averages can achieve rapid convergence of arbitrary polynomial order. However, more rigorous theoretical work is still lacking in this area. To this end, inspired by the fundamental work of Das and Yorke, the authors recently made interesting findings in [34, 35, 33]–namely that universally–or saying in terms of full measure, the weighted Birkhoff averages (as well as the Cesàro weighted averages) exhibit exponential convergence as long as the observables are analytic and irrespective of finite dimensions (quasi-periodic) or infinite dimensions (almost periodic). This comes somewhat unexpectedly since it can break through what is known as “curse of dimensionality”–a common phenomenon in high-dimensional systems where complexity grows exponentially with each additional dimension. More generally, the authors also discussed abstract scenarios admitting weaker nonresonance and regularity and present certain balancing criteria to achieve rapid convergence. Due to brevity considerations, these details are not elaborated upon here.

Based on these theoretical foundations, many researchers have utilized this weighting method to perform numerical calculations on various physical systems which has significantly increased their efficiency. On this aspect, see Calleja et al. [4], Das et al. [6], Sander and Meiss [27, 28, 29], Meiss and Sander [22], Duignan and Meiss [9], Blessing and Mireles James [3] among several others; these contributions provide strong evidence supporting rapid convergence. Specifically, Duignan and Meiss [9] employed this weighting method to distinguish between regular (in fact, quasi-periodic there) and chaotic systems. To be more precise, under the weighted Birkhoff averages via (1.1), regular systems typically demonstrate a rapid exponential convergence. In contrast, chaotic systems exhibit at most an one-order polynomial slow convergence. This raises an intriguing question: what other scenarios might these so-called regular systems applicable to weighted Birkhoff averages encompass?

The primary motivations behind this work are twofold: (M1) We aim to delve deeper into the practical applications of weighted Birkhoff averages beyond quasi-periodic and almost periodic systems which admit zero Lyapunov exponents. We believe at least that, for dynamical systems having negative maximal Lyapunov exponents, Birkhoff averages along trajectories, once weighted by the function (1.1), should also exhibit exponential convergence under certain generic conditions. (M2) Currently, the analysis of the convergence rates for the weighted Birkhoff averages with the weighting function (1.1) is almost entirely qualitative (at least, not particularly precise when compared with numerical simulations of the convergence rates). We suspect that this imprecision is due to limitations in previous techniques. Therefore, we aim to develop new approaches to provide a quantitative version of these convergence rates.

Based on the above considerations, we first start with some typical examples, namely exponentially decaying waves (e.g., $e^{-\lambda x}\sin(\rho x)$ and $e^{-\lambda x}\cos(\rho x)$ with certain $\lambda>0$ and $\rho\in\mathbb{R}$), which constitute some fundamental solutions among others for linear matrix differential equations described as $x^{\prime}=Ax$ (other solutions have more polynomial parts than these, depending on the multiplicity of the eigenvalues, see Section 6), where all eigenvalues of the constant matrix $A$ lie inside the unit circle $\mathbb{S}^{1}$. Such decaying waves can also be interpreted as projections onto planes from spiral waves within $3$-dimensional space. For further insights into the connections between decaying waves and dynamics systems, we refer readers to Liu et al. [20, 21], Wang et al. [37], Xing et al. [38] inclusive of their references which provide an exploration into this domain. As a main contribution, we demonstrate that Laskar’s weighted averaging method not only significantly accelerates the Birkhoff averages along quasi-periodic or almost periodic orbits, but also proves effective for other scenarios, involving weighted Birkhoff averages of decaying waves, with extremely rapid exponential convergence rates contrasting with the universal one-order polynomial type observed in unweighted cases (Theorem 1.1 and Remark 1.1). Specifically, our quantitative convergence rates prove to be remarkably precise, as evidenced by the numerical simulations illustrated in Figure 1 within Section 8. Furthermore, we proceed to discuss more general decaying waves in series forms and arrive at optimal results somewhat unexpectedly (Theorem 1.2).

Before delving into the main subject, let us first introduce some basic notations for the sake of simplicity. We define that in the limit process as $x\to+\infty$, the statements $f_{1}(x)=\mathcal{O}\left(f_{2}(x)\right)$ and $f_{1}(x)=\mathcal{O}^{\#}\left(f_{2}(x)\right)$ imply there exists a positive constant $c>0$, such that $|f_{1}(x)|\leqslant cf_{2}(x)$ and $c^{-1}f_{2}(x)\leqslant|f_{1}(x)|\leqslant cf_{2}(x)$, respectively. We also denote by $D^{n}f$ the $n$-th derivative of a sufficiently smooth function $f$. Additionally, for any fixed number $x\in\mathbb{R}$, we define ${\left\|x\right\|_{\mathbb{T}}}:={\inf_{n\in\mathbb{Z}}}\left|{x-2\pi n}\right|$ as the distance to the closest element of the set $2\pi\mathbb{Z}$, or equivalently, it represents the metric on the standard torus $\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}$ identified with $[0,2\pi)$.

In what follows, we proceed to present our main results. Consider the unweighted time average of the decaying wave as

as well as the $w$-weighted type given by

The first main result in this paper is stated as follows.

Theorem 1.1 provides an upper bound on the convergence rate of the weighted Birkhoff average ${{\rm{WDW}}_{N}\left({\lambda,\rho,\theta}\right)}$, which depends simultaneously on the decaying parameter $\lambda$ and the rotating parameter $\rho$. As we will demonstrate in the numerical simulation within Section 8, this estimate is remarkably precise.

The following conclusion concerning the averages of superimposed waves is a direct corollary of Theorem 1.1.

Indeed, when the decaying parameters $\lambda_{k}=\lambda>0$, Corollary 1.1 covers the cases of decaying quasi-periodic or decaying almost periodic. For a given set of rotation parameters $\{\rho_{k}\}_{k=1}^{d}$ with $1\leqslant d<+\infty$ ($=+\infty$), if for any $0\neq k\in\mathbb{Z}^{d}$, $\sum\nolimits_{j=1}^{d}{k_{j}\rho_{j}}\neq 0$, then the motion exhibited is quasi-periodic (almost periodic). See [33] for further characterization of these two concepts, as well as specific examples in Kozlov [15], for instance. However, the control coefficient in Theorem 1.1 may tend to $+\infty$ as the decaying parameter $\lambda$ tends to $0^{+}$. Therefore, if the case considered in Corollary 1.1 allows the decaying parameters $\lambda_{k}$ to tend to $0^{+}$ (acting similarly to Bessel functions), then with the variation in the arithmetic properties of the rotation parameters $\rho_{k}$, more complicated small denominator problems may arise. This needs to be dealt with in combination with the techniques in [33, 34, 35] (see also Section 7), which we prefer not to discuss here.

To enhance the applicability of our results, we present the quantitative Theorem 1.2 below. Prior to this, we denote by $A(\mathbb{T})$ the space consisting of continuous functions on $\mathbb{T}$ having an absolutely convergent Fourier series, i.e., the functions $f\left(x\right)=\sum\nolimits_{k\in\mathbb{Z}}{{{\hat{f}}_{k}}{e^{ikx}}}$ for which $\sum\nolimits_{k\in\mathbb{Z}}{|{{\hat{f}}_{k}}|}<+\infty$. With the norm defined by ${\left\|f\right\|_{A\left(\mathbb{T}\right)}}:=\sum\nolimits_{k\in\mathbb{Z}}{|{{\hat{f}}_{k}}|}$, ${A\left(\mathbb{T}\right)}$ is a Banach space isometric to $l^{1}$ with algebra property, see Katznelson [14] for instance. For $0<\alpha<1$, we denote by $C^{\alpha}(\mathbb{T})$ the space of $\alpha$-Hölder functions endowed with the norm

We also denote by $C^{n}(\mathbb{T})$ the space of functions on $\mathbb{T}$ with $n$-order continuous derivatives, where $n\in\mathbb{N}$.

Recall that in [22, 5, 33], achieving fast convergence of the weighted Birkhoff average necessitates strong regularity for the observable, e.g., the $C^{\infty}$ regularity or even analyticity. However, as shown for the decaying wave in (I), incorporating the decaying part compensates for the lack of regularity, thereby naturally achieving the exponential convergence. Moreover, the observable in (III) could admit very weak regularity, for example, discontinuity except of $x=0$.

The remainder of this paper is organized as follows. To highlight the ideas underlying our proof, we provide a detailed analysis of Theorem 1.1 in Section 2. Building upon this foundation, we give a more concise proof for Theorem 1.2 in Section 4. We also quantitatively discuss more general weighting functions in Section 5 and demonstrate their potential to lead exponential convergence of weighted Birkhoff averages, thereby providing theoretical validation for the numerical results by Das et al. [6], Duignan and Meiss [9], and Calleja et al. [4], among others. Although Theorems 1.1 and 1.2 consider discrete cases, we illustrate in Section 6 that such results extend naturally to continuous cases as well, and indeed, even simpler. In addition to the exponentially decaying waves addressed in Theorems 1.1 and 1.2, our newly developed techniques are also applicable to the weighted Birkhoff averages associated with more general types of decaying waves, especially orbits from certain nonlinear dynamical systems having negative maximum Lyapunov exponents, as detailed in Section 6. In Section 7, following careful approaches introduced in this paper, we revisit the exponential convergence rates of weighted Birkhoff averages along quasi-periodic orbits as initially discussed in [33], contributing an enhanced quantitative result (Theorem 7.1) to the existing research. As part of the main contributions, we show that:

• •

  (Part of Theorem 7.1) In the weighted Birkhoff average (1.2), for almost all quasi-periodic mappings $T$, whenever the observable is analytic, it converges to the spatial average $\int_{{\mathbb{T}^{d}}}{f\left(\theta\right)d\theta}$ at an exponential rate of $\mathcal{O}\left({\exp\left({-{c_{\rm I}}{N^{\frac{1}{{d+2}}}}{{\left({\ln N}\right)}^{-\frac{\zeta}{{d+2}}}}}\right)}\right)$ in the $C^{0}$ topology, where $\zeta>1$ is arbitrarily given, and ${c_{\rm I}}>0$ is some universal constant.

Finally, we present in Section 8 numerical simulation for a universal example which showcase the precision of our results, from a quantitative perspective.