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Dynamics of typical Baire-1 functions on the interval

Steele T. H.

Journal of Applied Analysis

2017

Vol. 23, nr 2

59--64

Let I = [0, 1], and let bB1 be the set of Baire-1 self-maps of I. For f ∈ bB1, let Λ(f) = ⋃x∈I ω(x, f) be the set of ω-limit points of f . We prove the following: - There exists a residual subset S of bB1 such that for any f ∈ S and x ∈ I the ω-limit set ω(x, f) is contained in the set of points at which f is continuous, and ω(x, f) is an∞-adic adding machine. - There exists a residual subset S of bB1 such that for any f ∈ S and for any ε > 0 there exists a natural number M such that fm (I) ⊂ Bε(Λ(f)) whenever m > M. Moreover, f : Λ(f) → Λ(f) is a bijection.

Global attractor for the convective Cahn-Hilliard equation in Hk

Zhao X., Duan N.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 1

53-64

We consider the convective Cahn-Hilliard equation with periodic boundary conditions. Based on the iteration technique for regularity estimates and the classical theorem on existence of a global attractor, we prove that the convective Cahn-Hilliard equation has a global attractor in Hk.