WITHDRAWN: Almost automorphy of surjective semiflows on compact Hausdorff spaces

Abstract

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...

The McMahon pseudo-metrics of minimal semiflows with invariant measures

“…Based on the recent work [12,5,9], we will generalize the above classical Theorem B to semiflows on compact T 2 -spaces; see Theorem 1.4 below. Definition 1.3.…”

“…Definition 1.13 (cf. [28,29,8,7] for T in groups and [9] for any semiflows). Let (T, X) be any semiflow.…”

“…Since an a.a. point is a distal point of any surjective (T, X) (cf. [9,Lemma 1.8] and [17, Theorem 9.13]), hence an a.a. surjective semiflow is point-distal and so minimal. We shall consider another Veech relation U(X) in §3.3.…”

“…Lemma 5.1 (cf. [9,Theorem 5.11]; also [29,Theorem 1.2] for T in abelian groups). Let (T, X) be an invertible minimal semiflow.…”