Let \(X\) be an infinite-dimensional separable Banach space. A linear and continuous operator \(T\in L(X)\) is said to be hypercyclic if there exists some \(x\in X\) whose orbit under \(T\) is dense. If, in addition, the set of periodic points for \(T\) is dense, then \(T\) is said to be chaotic. Analogous definitions can also be stated for a \(C_0\)-semigroup \(\{T_t\}_{t\geq 0}\) of linear and continuous operators in \(L(X)\).
There are some differences in the treatment of hypercyclicity and chaos: It is known that every infinite-dimensional separable Banach space supports a hypercyclic operator [see {S.I.Ansari}, J. Funct.Anal.148, No.2, 384–390 (1997; Zbl 0898.47019) and L.Bernal-González, Proc.Am.Math.Soc.127, No.4, 1003–1010 (1999; Zbl 0911.47020); see also J.Bonet and A.Peris, J.Funct.Anal.159, No.2, 587–595 (1998; Zbl 0926.47011) for the Fréchet case]. However, the counterpart for chaotic operators is not fulfilled [J.Bonet, F.Martínez-Giménez and A.Peris, Bull.Lond.Math.Soc.33, No.2, 196–198 (2001; Zbl 1046.47008)]. In both cases, analogous results can also be stated for \(C_0\)-semigroups [T.Bermúdez, A.Bonilla and A.Martinón, Proc.Am.Math.Soc.131, No.8, 2435–2441 (2003; Zbl 1044.47006)].
The hypercyclicity is preserved if we restrict ourselves to sub-semigroups: S.I.Ansari proved that if \(T\) is hypercyclic, then \(T^p\) is hypercyclic for every \(p\in\mathbb{N}\) [J. Funct.Anal.128, No.2, 374–383 (1995; Zbl 0853.47013)]. On the other hand, F.León-Saavedra and V.Müller proved that the operator \(T\) is hypercyclic if and only if \(\lambda T\) is hypercyclic for every \(\lambda\in\mathbb{C}\) with \(|\lambda|=1\) [Integral Equations Oper.Theory 50, No.3, 385–391 (2004; Zbl 1079.47013)]. The ideas in this paper gave the key to prove that all nontrivial operators on a hypercyclic \(C_0\)-semigroup \(\{T_t\}_{t\geq 0}\) are also hypercyclic [J.A.Conejero, V.Müller and A.Peris, J. Funct.Anal.244, No.1, 342–348 (2007; Zbl 1123.47010)].
In the paper under review, the authors give a negative answer to the problem of extending these last two results to the chaotic setting. This gives a further insight of the relations between the dynamical properties of a \(C_0\)-semigroup and its operators.
The authors give an example of a chaotic operator \(T\) such that \(\lambda T\) is not chaotic for certain \(\lambda\in\mathbb{C}\) with \(|\lambda|=1\). Besides, they give an example of a chaotic \(C_0\)-semigroup \(\{T_t\}_{t\geq 0}\) such that there exists \(t_0,t_1\neq 0\) such that \(T_{t_0}\) is chaotic and \(T_{t_1}\) is not chaotic. Even more, they construct a chaotic \(C_0\)-semigroup that does not contain any chaotic operator. The constructions are given from a detailed study of the point spectrum of chaotic operators.