Isospectral theory of the Lax pairs of both 3D and 2D Euler equations of inviscid fluids is developed. Eigenfunctions are represented through an ergodic integral. The Koopman group and mean ergodic theorem are utilized. Further harmonic analysis results on the ergodic integral are introduced. The ergodic integral is a limit of the oscillatory integral of the first kind.

We prove an L subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of sparser universally L-good sequences than had been previously established. We extend this theorem to a more general setting of ergodic group actions.

An entangled ergodic theorem was introduced in [1] in connection with the quantum central limit theorem, and clearly formulated in [6]. Namely, let U be a unitary operator on the Hilbert space H, and for m ≥ k, α : {1, . . . , m} 7→ {1, . . . , k} a partition of the set {1, . . . , m} in k parts. The entangled ergodic theorem concerns the convergence in the strong, or merely weak (s–limit, or w...

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem. Let (X,B, μ) be a probability space, T : X → X a (possibly noninvertible) measurepreserving transformation, and f ∈ L(X,B, μ). Let

This is an earlier, but more general, version of ”An L Ergodic Theorem for Sparse Random Subsequences”. We prove an L ergodic theorem for averages defined by independent random selector variables, in a setting of general measure-preserving group actions. A far more readable version of this paper is in the works.

By exploiting the Denjoy theorem in topological dynamics and the unique ergodic theorem in ergodic theory, we will give a classification of all solutions of asymmetric p-Laplacian oscillators with periodic coefficients. AMS (MOS) Subject Classification. Primary: 34D08; Secondary: 37E10, 37A25.

We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff’s pointwise ergodic theorem for ergodic transformations, are stable in this...

We offer a proof of the following nonconventional ergodic theorem: Theorem. If Ti : Z y (X,Σ, μ) for i = 1, 2, . . . , d are commuting probability-preserving Z-actions, (IN )N≥1 is a Følner sequence of subsets of Z, (aN )N≥1 is a base-point sequence in Z and f1, f2, . . . , fd ∈ L∞(μ) then the nonconventional ergodic averages