相关论文: Uniform convergence rate for Birkhoff means of cer…

For an ergodic flow, a range of rates of convergence of Birkhoff averages from the maximum rate to an arbitrarily slow rate is realized by choosing the averaging function. For torus windings, the continuity of the averaging functions is…

In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation…

The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages, $\Sigma_{n=1}^N f(x_n)/N$ of a function $f$ along an ergodic trajectory $(x_n)$ of a function $T$ converges to the space average $\int f d\mu$, where…

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, $\Sigma_{n=0}^{N-1} f(x_n)/N$ of a function $f$ along a length $N$ ergodic trajectory $(x_n)$ of a function $T$ converge to the space average $\int f d\mu$,…

In this paper, we study the topological spectrum of weighted Birkhoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave…

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…

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an…

Let T be an ergodic automorphism of the d-dimensional torus. In the spirit of Le Borgne, we give conditions on the Fourier coeffi cients of a real valued function f under which the Birkhoff sums satis fy a strong invariance principle. Next,…

We study a skew product with a curve of neutral points. We show that there exists a unique absolutely continuous invariant probability measure, and that the Birkhoff averages of a sufficiently smooth observable converge to a normal law or a…

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…

Fixed a continuous kernel K on the $d$-dimensional torus, we consider a generalization of the univariate $sk$-spline to the torus, associated with the kernel K. It is proved an estimate which provides the rate of convergence of a given…

The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff averages (i.e. time averages computed along a trajectory) converge to the space average. For sufficiently smooth systems, our small modification of numerical Birkhoff…

It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost…

In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic…

We study very smooth functions on the real line, namely Schwartz functions, that satisfy a finite identity relating their translates and a single modulation. Concretely, we assume there is a nontrivial linear combination of translates of…

For $\mathbb{R}^d$-ergodic means constructed over a strictly convex set, a spectral criterion for homogeneous rates of convergence is obtained. From Hertz's result on the asymptotics of the Fourier transform of the indicator of a strictly…

We study Birkhoff sums as distributions. We obtain regularity results on such distributions for various dynamical systems with hyperbolicity, as hyperbolic linear maps on the torus and piecewise expanding maps on the interval. We also give…

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to…

In this paper, we develop numerical methods based on the weighted Birkhoff average for studying two-dimensional invariant tori for volume-preserving maps. The methods do not rely on symmetries, such as time-reversal symmetry, nor on…

Von Neumann's original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to…