arXiv subjects

Zhicheng Tong

Publications and source records attributed to Zhicheng Tong.

18 recordsLinked to original sources

arXiv2026

Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations

In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that no admissible choice of weighting function can, in general, overcome the lower bounds imposed by this regularity. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.

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arXiv2026

Complete characterization of planar $C^1$ linearizability and optimal conjugacy regularity

We establish a complete characterization of local $C^1$ linearizability for planar diffeomorphisms. For every prescribed hyperbolic real Jordan form, we obtain necessary and sufficient integrability conditions on the modulus of continuity of the derivative for local $C^1$ linearizability. These criteria reduce to three canonical thresholds: the classical, polynomially weighted, and squared-logarithmically weighted Dini conditions. Each threshold is optimal: whenever the corresponding condition fails, we construct a diffeomorphism with the identical linear part that is not locally $C^1$ linearizable. For every non-hyperbolic linear part, explicit polynomial counterexamples show that smoothness alone cannot guarantee even local topological linearization. Beyond $C^1$ existence, we determine the optimal regularity for the derivative of the linearizing conjugacy in every hyperbolic case, with the optimality established by counterexamples.

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arXiv2026

Laskar's frequency map analysis revisited

In this paper, we establish the exponential convergence of Laskar's pioneering frequency map analysis, strictly improving upon classical polynomial bounds. By utilizing appropriately chosen weighting functions, we achieve such exponential rates in the analytic quasi-periodic regime for frequency vectors satisfying Diophantine or Brjuno nonresonance conditions. Furthermore, we extend the theoretical framework beyond the analytic quasi-periodic regime into the analytic almost periodic setting. By employing and generalizing Bourgain's framework, we accommodate a much broader class of anisotropic spatial structures. These results yield the first unified theory of exponential convergence for frequency map analysis, revealing the interplay among analyticity, spatial structures, nonresonance conditions, the choice of weighting functions, and convergence rates. This framework has important implications for applications in fields including celestial mechanics. Moreover, the novel techniques developed herein yield new insights into the study of weighted Birkhoff averages.

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arXiv2026

Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrary nonresonance and low regularity

Since Krengel's work [Kre78] in 1978, it has been widely known that no effective rate of convergence exists in the ergodic theorem. For toral translations, however, by choosing appropriate weights one can accelerate the convergence of ergodic averages to an exponential rate, but this intuitively requires both highly nonresonant frequencies and very regular observables. In this paper, we uncover a new phenomenon: even for any given nonresonant frequency, there exists a non-trivial family of weights and observables of low regularity such that the weighted Birkhoff averages along quasi-periodic orbits converge at a quantitative, uniform, and exponential rate. This not only yields a finer understanding of the deep interaction between nonresonance and regularity in ergodic theory, but also stands as a weighted counterpart to a Yoccoz-type result [Yoc80,Yoc95].

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arXiv2026

Sharp regularity of a weighted Sobolev space over $ \mathbb{T}^n $ and its relation to finitely differentiable KAM theory

In this paper, we investigate the sharp regularity properties of a special weighted Sobolev space defined on the $ n $-dimensional torus, which is of independent interest. As a key application, we show that for almost all $ n $-dimensional vector fields, the Kolmogorov-Arnold-Moser (KAM) theory holds via this regularity, and in this case, the perturbation must have classical derivatives up to order $ \left[ {n/2} \right] $, yet it can admit unbounded weak derivatives from order $ \left[ {n/2} \right]+1 $ to $ n$. This result may appear surprising within the classical framework of KAM theory. We also provide further discussion of historical KAM theorems and relevant counterexamples. These findings constitute a new step in the long-standing KAM regularity conjecture.

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arXiv2026

An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers

In this paper, we present two infinite-dimensional Kolmogorov theorems based on non-resonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a Legendre-type nondegeneracy condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with a universally prescribed frequency independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \[\frac{{{{\rm d}^2}{x_n}}}{{{\rm d}{t^2}}} + V'\left( {x_n} \right) = \varepsilon_n {W'\left( {x_{n + 1} - {x_n}} \right) - \varepsilon_{n-1}W'\left( {{x_n} - {x_{n - 1}}} \right)} ,\quad n \in \mathbb{Z},\] or even for more general perturbed networks, frequency-preserving almost periodic breathers do persist, provided that the local potential $ V $ and the coupling potential $ W $ satisfy certain assumptions. In particular, this yields the first frequency-preserving result for the Aubry--MacKay conjecture [MA94,Aub95].

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arXiv2026

Relation between irrationality and regularity for $ C^1 $ conjugacy of $ C^2 $ circle diffeomorphisms to rigid rotations

By introducing the modulus of continuity, we first establish the corresponding cross-ratio distortion estimates under $ C^2 $ smoothness, and further derive a Denjoy-type inequality, which is almost optimal for dealing with circle diffeomorphisms. The latter plays a prominent role in the study of $ C^1 $ conjugacy to irrational rotations. We also establish an explicit integrability correlation between continuity and irrationality for the first time. Furthermore, the regularity of the conjugation is addressed and proved to be sharp.

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arXiv2026

Weighted Birkhoff averages: Deterministic and probabilistic perspectives

In this paper, we survey physically related applications of a class of weighted quasi-Monte Carlo methods from a theoretical, deterministic perspective, and establish quantitative universal rapid convergence results via various regularity assumptions. Specifically, we introduce weighting with compact support to the Birkhoff ergodic averages of quasi-periodic, almost periodic, and periodic systems, thereby achieving universal rapid convergence, including both arbitrary polynomial and exponential types. This is in stark contrast to the typically slow convergence in classical ergodic theory. As new contributions, we not only discuss more general weighting functions but also provide quantitative improvements to existing results; the explicit regularity settings facilitate the application of these methods to specific problems. We also revisit the physically related problems and, for the first time, establish universal exponential convergence results for the weighted computation of Fourier coefficients, in both finite-dimensional and infinite-dimensional cases. In addition to the above, we explore results from a probabilistic perspective, including the weighted strong law of large numbers and the weighted central limit theorem, by building upon the historical results.

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arXiv2025

Weighted multiple ergodic averages via analytic observables over $ \mathbb{T}^\infty $: Is exponential pointwise convergence universal?

By employing an accelerated weighting method, we establish arbitrary polynomial and exponential pointwise convergence for multiple ergodic averages under general balancing conditions in both discrete and continuous settings, including quasi-periodic and almost periodic cases. We also present joint Diophantine rotations as explicit applications. Specifically, for the first time, by excluding nearly rational rotations with zero measure, we address the fundamental question of whether exponential pointwise convergence via analytic observables is universal, even when multiplicatively averaging over the infinite-dimensional torus $ \mathbb{T}^\infty $. We achieve this by introducing an innovative approach that effectively overcomes the previous difficulties. Moreover, by constructing counterexamples concerning multiple ergodicity, we highlight the indispensability of the joint nonresonance and establish the optimality of our weighting method in preserving rapid convergence. We also provide numerical simulations and analysis to further illustrate and validate our results.

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arXiv2025

Towards sharp regularity: Full-dimensional tori in $ C^\infty $ vector fields over $ \mathbb{T}^\infty $

We consider the $ C^1 $ linearization of a perturbed vector field $ ω+P $ over the infinite-dimensional torus $ \mathbb{T}^\infty $, and determine the sharp regularity requirement of the perturbation $ P $ conjugating the unperturbed one $ ω$ onto $ ω-\tildeω+P $ via a small modifying term $ \tildeω $. We discuss the Diophantine type introduced by Bourgain, investigate the universal nonresonance, and provide the weakest known regularity of perturbations for which KAM applies. Our results allow for lower regularity than analyticity such as Gevrey regularity or even $ C^\infty $ regularity. We propose a new KAM scheme with a balancing sequence to overcome the non-polynomial nonresonance, differing from the usual Newtonian approach. Thereby, in addition to deriving the sharp Gevrey exponent along Diophantine nonresonance, we answer the fundamental question of what the minimum regularity required for KAM is in the infinite-dimensional setting: $ C^\infty $ regularity. Our linearization technique also applies to the quasi-periodic case over $ \mathbb{T}^n $.

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arXiv2025

A generic approach via relative singularity and controllability: Frequency-preserving with arbitrarily weak regularity in parameterized Hamiltonian systems

In this paper, we introduce a novel and generic approach to prove the persistence of frequency-preserving invariant tori in parameterized Hamiltonian systems, addressing irregular continuity with respect to parameters. Unlike traditional methods that strongly rely on domain extraction techniques or uniform weak convexity of the frequency mapping, we propose the concepts of relative singularity and controllability for the first time. These concepts enable us to deal with a wide range of explicit parameterized Hamiltonian systems with arbitrarily weak regularity, thereby overcoming a previously insurmountable challenge. We also construct several counterexamples to highlight the indispensability of our new conditions in the sense of frequency-preserving. Furthermore, we demonstrate the broad applicability of our results to various cases with explicit arbitrarily weak regularity, including the partial frequency-preserving case and the infinite-dimensional case without any spectral asymptotics. Overall, our approach, based on the concepts of relative singularity and controllability, illustrates its genericity in the frequency-preserving KAM theory.

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arXiv2025

Quantitative uniform exponential acceleration of averages along decaying waves

In this study, utilizing a specific exponential weighting function, we investigate the uniform exponential convergence of weighted Birkhoff averages along decaying waves and delve into several related variants. A key distinction from traditional scenarios is evident here: despite reduced regularity in observables, our method still maintains exponential convergence. In particular, we develop new techniques that yield very precise rates of exponential convergence, as evidenced by numerical simulations. Furthermore, this innovative approach extends to quantitative analyses involving different weighting functions employed by others, surpassing the limitations inherent in prior research. It also enhances the exponential convergence rates of weighted Birkhoff averages along quasi-periodic orbits via analytic observables. To the best of our knowledge, this is the first result on the uniform exponential acceleration beyond averages along quasi-periodic or almost periodic orbits, particularly from a quantitative perspective.

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arXiv2024

Full-dimensional KAM torus with frequency-preserving in infinite-dimensional Hamiltonian systems

In this paper, we present two infinite-dimensional KAM theorems with frequency-preserving for a nonresonant frequency of Diophantine type or even weaker. To be more precise, under a nondegenerate condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with the specified frequency independent of any spectral asymptotics, by advantage of the generating function method. This appears to be the first Kolmogorov type result in the infinite-dimensional context. As a direct application, we provide a positive answer to Bourgain's conjecture: full-dimensional invariant tori for 1D nonlinear Schrödinger equations do exist.

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arXiv2023

Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity

Beyond Hölder's type, this paper mainly concerns the persistence and remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allows the non-integrable part being critical finitely smooth. To achieve this goal, besides investigating the Jackson approximation theorem towards only modulus of continuity, we demonstrate an abstract regularity theorem adapting to the new iterative scheme. Via these tools, we obtain a KAM theorem with sharp differentiability hypotheses, asserting that the persistent torus keeps prescribed universal Diophantine frequency unchanged. Further, the non-Hölder regularity for invariant KAM torus as well as the conjugation is explicitly shown by introducing asymptotic analysis. To our knowledge, this is the first approach to KAM on these aspects in a continuous sense, and we also provide two systems, which cannot be studied by previous KAM but by ours.

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arXiv2023

Moser's Theorem with Frequency-preserving

This paper mainly concerns the KAM persistence of the mapping $\mathscr{F}:\mathbb{T}^{n}\times E\rightarrow \mathbb{T}^{n}\times \mathbb{R}^{n}$ with intersection property, where $E\subset \mathbb{R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser type results about the invariant torus of mapping $\mathscr{F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $ \mathbb{T}^n $, it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even Hölder type. We also touch the parameter without dimension limitation problem under such settings.

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arXiv2023

Universal frequency-preserving KAM persistence via modulus of continuity

In this paper, we study the persistence and remaining regularity of KAM invariant torus under sufficiently small perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus of continuity, as a generalization of classical Hölder continuous circumstances. To achieve this goal, we extend the Jackson approximation theorem to the case of modulus of continuity, and establish a corresponding regularity theorem adapting to the new iterative scheme. Via these tools, we establish a KAM theorem with sharp differentiability hypotheses, which asserts that the persistent torus keeps prescribed universal Diophantine frequency unchanged and reaches the regularity for persistent KAM torus beyond Hölder's type.

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arXiv2023

KAM theorem on modulus of continuity about parameter

In this paper, we study the Hamiltonian systems $ H\left( {y,x,ξ,\varepsilon } \right) = \left\langle {ω\left( ξ\right),y} \right\rangle + \varepsilon P\left( {y,x,ξ,\varepsilon } \right) $, where $ ω$ and $ P $ are continuous about $ ξ$. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain transversality condition and weak convexity condition for the frequency mapping $ ω$. As a direct application, we prove a KAM theorem when the perturbation $P$ holds arbitrary Hölder continuity with respect to parameter $ ξ$. The infinite dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in parameter beyond Hölder's type.

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arXiv2022

Exponential convergence of weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rate of weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

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