The notion of distributional chaos has been recently added to the study of the linear dynamics of operators and -semigroups of operators. We will study this notion of chaos for some examples of -semigroups that are already known to be Devaney chaotic. 1. Introduction During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of -semigroups of linear and continuous operators. The recent monograph by Grosse-Erdmann and Peris Manguillot [1] is a good reference for researchers interested in the study of linear dynamics. In particular, it contains a chapter dedicated to analyze the dynamics of -semigroups. See also [2], which contains additional information on further topics in the area. In the sequel, let be an infinite-dimensional separable Banach space. A -semigroup is a family of linear and continuous operators such that , for all , and for all , we have pointwise on . We say that a -semigroup is Devaney chaotic if it is transitive and it has a dense set of periodic points. On the one hand, a -semigroup is transitive if for any pair of nonempty open sets there is some such that . In this setting, transitivity is equivalent to the existence of some with dense orbit in , that is, , see for instance [1, Th. 1.57]. This phenomenon is usually known in operator theory as hypercyclicity, and such a vector is said to be a hypercyclic vector for . On the other hand, a vector is said to be a periodic point for if there is some such that . Other definitions of chaos, such as the one introduced by Li-Yorke and the one of distributional chaos introduced by Schweizer and Smítal, have been also considered. The relationships between these two notions in the Banach space setting have been recently studied in [3]. We recall that a -semigroup is said to be Li-Yorke chaotic if there exists an uncountable subset , called the scrambled set, such that for every pair of distinct points, we have that Clearly, every hypercyclic -semigroup is Li-Yorke chaotic: we just have to fix a hypercyclic vector and consider as a scrambled set, as it is indicated in [4, page 84]. Distributional chaos is inspired by the notion of Li-Yorke chaos. In order to define it, given a subset , we define its upper density as , where stands for the Lebesgue measure on .
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