Semigroup Forum (2018) 96:596–599
Indecomposability and Devaney’s chaoticity
of semiﬂows with an arbitrary acting abelian
Received: 20 March 2017 / Accepted: 18 June 2017 / Published online: 21 June 2017
© Springer Science+Business Media, LLC 2017
Abstract We prove that a semiﬂow with an arbitrary acting abelian topological semi-
group is Devaney chaotic if and only if it is nonminimal, indecomposable and has a
dense set of periodic points. This generalizes a result of X. Wang and Y. Huang.
Keywords Semiﬂow · Devaney’s chaos · Indecomposability · Periodic points ·
Sensitivity · Topological transitivity
1 Notation and preliminaries
We assume that T is an arbitrary abelian unital topological semigroup (i.e., an arbitrary
abelian topological monoid) and (X, d) is a metric space. A triple (T , X,π), where
π : T × X → X is a jointly continuous monoid action of T on X is called a semiﬂow
and is usually denoted shortly by (T , X ). The element π(t, x) will be denoted by t.x
or tx, so that the deﬁning conditions for a semiﬂow have the form
s.(t.x) = (s + t). x,
0.x = x,
Communicated by Jimmie D. Lawson.
The author was partially supported by the National Science Foundation Grant DMS-1405815.
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA