Relationships among Various Chaos for Linear Semiflows Indexed with Complex Sectors

Abstract

In this paper, we investigate the relationships among point transitivity, topological transitivity, Li–Yorke chaos, and the existence of irregular vectors for a linear semiflow {Tt}t∈Δ indexed with a complex sector. We reveal the equivalence between topological transitivity and point transitivity for a linear semiflow {Tt}t∈Δ, especially in case the range of some operator Tt,t∈Δ is not dense. We also prove that Li–Yorke chaos is equivalent to the existence of a semi-irregular vector and that point transitivity is stronger than the existence of an irregular vector for any linear semiflow Ttt∈Δ. At last, unlike the conclusion for traditional linear dynamical systems, we show that there exists a Li–Yorke chaotic C0-semigroup Ttt∈Δ without irregular vectors. The results and proof methods presented in this paper demonstrate the differences in the dynamical behavior between linear semiflows {Tt}t∈Δ and traditional linear systems with the acting semigroup S=Z+ and S=R+.

Keywords

• linear semiflows
• complex sector
• transitivity
• Li–Yorke chaos
• irregular vector