Almost-periodic dissipative actions of semigroups (Q912991)

The paper extends the results of \textit{Yu. I. Lyubich} [Ukr. Mat. Zh. 40, No.1, 70-74 (1988)] where the almost periodic representations of abelian topological semigroups S (with unit) operating in a metric compact space X were studied. Here is the commutativity of S no longer assumed and one let X be any complete metric space. For the abelian case additional results are obtained. For every \(s\in S\) let a(s) be a mapping \(X\to X\) such that (s,x)\(\mapsto a(s)x\) (x\(\in X)\) is separately continuous, \(a(e)=id\) and \(a(st)=a(s)a(t)\). The representation a of S in X is called almost periodic (a.p.) iff all orbits \(O(x)=\{a(s)x\}_{s\in S}\) are relatively compact. One assumes a dissipative, i.e. such that the mappings a(s) do not augment distances. Let \(\beta_ a\) be the closure of \(\{a(s)\}_{s\in S}\) in the space \(X^ X\) equipped with the topology of pointwise convergence. Then a is a.p. iff \(\beta_ a\) is compact. \(\beta_ a\) is then called Bohr compactification for a. It is a semigroup. If S is a group then so is \(\beta_ a\). The smallest two- sided ideal in a semigroup S (if it exists) is called its Sushkevich kernel k(S). A dissipative a.p. representation of S is called elementary if \(k(\beta_ a)\) is a group (this is always the case if S is abelian). In such case the invariant space \(\cap_{b\in \beta_ a}b(X)\) is called the support of a. The author proves that supp a is a closed set which is connected if X is connected. Further results are e.g.: the restricted function \(a| \sup p a\) is an isometry and supp a is the largest invariant space having this property. Moreover, all points of supp a are recurrent i.e. such that \(x\in \cap_{b\in \beta_ a}b(O(x))\). If S is abelian one can make S a directed set by putting \(s\geq t\) iff \(s=tu\) for some u. In this sense one can speak on \(\omega\)-limit points of an orbit O(x). This allows to introduce the notion of asymptotic almost periodicity (a.a.p.), in general weaker than a.p. (compare the following review Zbl 0699.22006). If one assumes the representation a a.a.p. instead of a.p. the above results remain valid. Moreover the set \(\Omega\) of all limit points of all orbits appears to be an attractor, i.e. \(\lim_{s} d(a(s)x,\Omega)=0\) for every \(x\in X\).