A Relation Between Almost Automorphic and Levitan Almost Periodic Points in Compact Minimal Flows

Miller, Alica

doi:10.1007/s10884-007-9085-y

Cited by 6 publications

(3 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then fundamental properties of a.a. functions on groups and a.a. flows on compact T 2 spaces have been systematically studied by Veech (cf. [28,29,31]) and then others (see, e.g., [23,13,27,26,24,4,3] and so on).…”

Section: Cmentioning

confidence: 97%

WITHDRAWN: Almost automorphy of surjective semiflows on compact Hausdorff spaces

Dai

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Let (T, X) with phase mapping (t, x) → tx be a semiflow on a compact T 2 -space X with phase semigroup T such that tX = X for each t of T . An x ∈ X is called an a.a. point if t n x → y, x ′ n → x ′ and t n x ′ n = y implies x = x ′ for every net {t n } in T . In this paper, we study the a.a. dynamics of (T, X); and moreover, we present a complete proof of Veech's structure theorem for a.a. flows.Keywords: Semiflow; almost automorphy; locally almost periodic; almost C-semigroup.2010 MSC: 37B05 · 37B20 · 20M20 0. Introduction Standing terminology. Throughout, unless specified otherwise, we assume:1. X is a non-empty compact T 2 space with the compatible symmetric uniform structure U X .In fact, U X is exactly the family of the symmetric open neighborhoods of ∆ X in the product space X × X.A. We will say that a jointly continuous map (t, x) → tx of T × X to X is the phase mapping of a semiflow with phase space X and with phase semigroup T , denoted (T, X), if ex = x for all x ∈ X and t(sx) = (ts)x for all s, t ∈ T, x ∈ X. Here (T, X) will be called a flow with phase group T when T is a group. Based on (T, X), T x = {tx | t ∈ T } is called the orbit or motion with initial state x ∈ X.B. As usual, (T, X) will be called a minimal semiflow if and only if cls X T x = X for all x ∈ X; and we could similarly define minimal subsets of (T, X). See, e.g., [15,13,10, 2]. Since X is compact here, the minimality will be equivalently described by the almost periodicity later.In this paper, we shall be mostly concerned with a kind of dynamical system-"invertible semiflow" which lies between flow and semiflow.). Let (T, X) be a semiflow. Then:, is a right-action semiflow [2, Lemma 0.6], which is called the reflection of (T, X).Recall that T is called a "right C-semigroup" if and only if cls T (T \ T t) is compact in T for all t in T (cf. [21, 2]). Now we will introduce a kind of more general topological semigroup than C-semigroup.We could similarly define almost left C-semigroup.3. If T is not only an almost right C-semigroup but also an almost left C-semigroup, then it is referred to as an almost C-semigroup.Clearly every topological group is a right C-semigroup and any right C-semigroup is an almost right C-semigroup like (R + , +) and (Z + , +) with the usual topologies. In fact, there are the following important almost right C-semigroups which are not right C-semigroups.Examples 0.5. We now construct some almost C-semigroups which are not C-semigroups.1. Let R n×n be the space of all real n × n matrices endowed with the usual topology. Since the nonsingular matrices are open dense in R n×n , thus T = (R n×n , •) is an almost C-semigroup. 2. Let M n be a compact boundaryless Riemannian manifold of dimension n and let T be the semigroup of C 0endomorphisms of M n with the C 0 -topology. Since Diff 1 (M n ) is dense in T , thus T is an almost C-semigroup under the composition of maps. 3. It is known that Diff 1 (M n ) is open in C 1 (M n , M n ) under the C 1 -topology. Let T be the C 1 -closure of Diff 1 (M n ) in C 1 (M n , M n...

show abstract

Section: Cmentioning

confidence: 97%

WITHDRAWN: Almost automorphy of surjective semiflows on compact Hausdorff spaces

Dai

2019

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…Fundamental properties of almost automorphic functions on groups and abstract almost automorphic minimal flows were studied by W.A. Veech [48,49] and others ( [3,14,23,25,[34][35][36]45,52] and the references therein). The almost automorphic solutions of differential equations were investigated in a series of recent works, see [5,12,15,21,35,37,45,55] and the references therein.…”

Section: Theorem 13 (Seementioning

confidence: 99%

“…63], Reich [39], Yi [45, p. 18], and the references therein). A relation between almost automorphic and N-almost periodic dynamics has been discussed [34]. In some papers, e.g.…”

Section: Theorem 13 (Seementioning

confidence: 99%