• Korean Journal of Mathematics
• /
• v.11 no.2
• /
• pp.161-167
• /
• 2003

The purpose of this paper is to study and characterize the notions of characteristic 0 and weakly almost periodicity in flows. In particular, we give sufficient conditions for the weakly almost periodic flow to be almost periodic.

• Journal of the Korean Mathematical Society
• /
• v.56 no.1
• /
• pp.39-52
• /
• 2019

The aim of this paper is to introduce the notions of (quasi) weakly almost periodic point, measure center and minimal center of attraction of amenable group actions, explore the connections of levels of the orbit's topological structure of (quasi) weakly almost periodic points and study chaotic dynamics of transitive systems with full measure centers. Actually, we showed that weakly almost periodic points and quasiweakly almost periodic points have distinct orbit's topological structure and proved that there exists at least countable Li-Yorke pairs if the system contains a proper (quasi) weakly almost periodic point and that a transitive but not minimal system with a full measure center is strongly ergodically chaotic.

• Communications of the Korean Mathematical Society
• /
• v.13 no.1
• /
• pp.123-129
• /
• 1998

In this paper, we study some dynamical properties of a weakly almost periodic flow. In particular we get, in a weakly almost periodic flow (X,T), the groups I and A(I) of all automorphisms of I are isomorphic, where E(X) is the enveloping semigroup of (X,T) and I is the minimal right ideal in E(X).

• Journal of the Korean Mathematical Society
• /
• v.59 no.6
• /
• pp.1229-1254
• /
• 2022

In this paper, we introduce the concepts of (quasi-)weakly almost periodic point and minimal center of attraction for ℤ^d₊-actions, explore the connections of levels of the topological structure the orbits of (quasi-)weakly almost periodic points and discuss the relations between (quasi-)weakly almost periodic point and minimal center of attraction. Especially, we investigate the chaotic dynamics near or inside the minimal center of attraction of a point in the cases of S-generic setting and non S-generic setting, respectively. Actually, we show that weakly almost periodic points and quasi-weakly almost periodic points have distinct topological structures of the orbits and we prove that if the minimal center of attraction of a point is non S-generic, then there exist certain Li-Yorke chaotic properties inside the involved minimal center of attraction and sensitivity near the involved minimal center of attraction; if the minimal center of attraction of a point is S-generic, then there exist stronger Li-Yorke chaotic (Auslander-Yorke chaotic) dynamics and sensitivity (ℵ₀-sensitivity) in the involved minimal center of attraction.

• Journal of the Korean Statistical Society
• /
• v.3 no.1
• /
• pp.13-16
• /
• 1974

In my former paper [3] I defined an almost periodicity of weakly sationary random processes (a.p.w.s.p.) and presented some basic results of it. In this paper I shall present some notes on the Fourier series of an a.p.w.s.p., resulting from [3]. All the conditions at the introduction of [3] are assumed to hold without repreating them here. The essential facts are as follows : The weakly stationary process $X(t,\omega), t\in(-\infty,\infty), \omega\in\Omega$, defined on a probability space $(\Omega,a,P)$, has a spectral representation $$X(t,\omega)=\int_{-\infty}^{infty}{e^{it\lambda\xi}(d\lambda,\omega)},$$ where $\xi(\lambda)$ is a random measure. Then, the continuous covariance $\rho(\mu) = E(X(t+u) X(t))$ has the form $$\rho(u)=\int_{-\infty}^{infty}{e^{iu\lambda}F(d\lambda)},$$ $E$\mid$\xi(\lambda+0)-\xi(\lambda-0)$\mid$^2 = F(\lambda+0) - F(\lambda-0) \lambda\in(-\infty,\infty)$, assumimg that $\rho(u)$ is a uniformly almost periodic function.

• Korean Journal of Mathematics
• /
• v.13 no.2
• /
• pp.235-239
• /
• 2005

In this paper we study some characterizations of proximal, distal and almost one to one homomorphisms of flows. In particular we show that if the almost one to one proximal extension of a minimal flow is weakly almost periodic, then it is minimal.