Generic distributional chaos and principal measure in linear dynamics
Generic distributional chaos and principal measure in linear dynamics are investigated. Sufficient conditions for a $C_0$-semigroup of operators on a Fréchet space to be generically distributionally chaotic are provided and applied to concrete examples. Furthermore, the distributionally chaotic dynamics of product operators (product $C_0$-semigroups, respectively) are considered. It is shown that under certain conditions, the product operator is generically distributionally chaotic if and only if there is a factor operator exhibiting generic distributional chaos. Another interesting finding is that there exist distributionally chaotic (but not hypercyclic) operators whose principal measure could be less than any fixed positive number.