In this study, we prove that a countably infinite number of one-parameterized one-dimensional dynamical systems preserve the Lebesgue measure and are ergodic for measure. The consider connect parameter region in which exact one almost all orbits diverge to infinity correspond critical points weak chaos tends occur (the Lyapunov exponent converging zero). These results generalization work by Adl...

We construct an invariant measure $\mu$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in support of are initial conditions global, unique solutions SQG, depend continuously on data. In addition, we is infinite dimensional, meaning it not locally a subset any compact set with finite Hausdorff dimension. Also, there global have arbitrarily large condition. Th...

In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found and is utilized to prove an ‘‘unstable disk theorem’’ through a simple energy–C...

We consider the phenomenon of concentration measures, which is restricted to case families compact connected Lie groups. While in literature, powerful general results regarding existence and its relations extremal amenability infinite dimensional groups have been determined, there are few explicit examples, specially determination region where measure concentrates. Since they can be relevant fo...