B. Schweizer and J. Smítal introduced distributional chaos and principal measures in [Trans. Am. Math. Soc. 344, No. 2, 737–754 (1994; Zbl 0812.58062)]. A systematic investigation of distributional chaos for linear operators on Fréchet spaces was undertaken by Bernardes, Bonilla, Müller and Peris in [N. C. Bernardes jun. et al., J. Funct. Anal. 265, No. 9, 2143–2163 (2013; Zbl 1302.47014)]. Later on, several authors developed the theory of distributional chaos for \(C_0\)-semigroups of operators on Banach and Fréchet spaces (see the references in the article under review).
The authors deal in their paper with the notion of generic distributional chaos for \(C_0\)-semigroups of operators on Fréchet spaces. This means that the set of all distributionally chaotic pairs for a \(C_0\)-semigroup forms a residual set. Not every distributionally chaotic operator (or \(C_0\)-semigroup) has this property. The authors present some sufficient conditions for a \(C_0\)-semigroup of operators on a Fréchet space to be generically distributionally chaotic. Then they focus on a concrete example of a \(C_0\)-semigroup on the space of real-valued continuous functions on \([0,\infty[\), which is proved to be Devaney chaotic, topologically mixing, and generically distributionally chaotic with principal measure \(1\).
Furthermore, they consider distributionally chaotic dynamics and principal measures of product operators and product \(C_0\)-semigroups. They show that, under certain conditions, the product operator is generically distributionally chaotic if and only if there is a factor operator exhibiting generic distributional chaos. Every distributionally chaotic operator and every \(C_0\)-semigroup on a Banach space has principal measure 1. It is interesting that the authors show that there are distributionally chaotic operators whose principal measure is less than any given positive number. Moreover, such operators need not be hypercyclic and hence may not be Devaney chaotic.