Let be an infinite-dimensional separable Banach space. A linear and continuous operator is said to be hypercyclic if there exists some whose orbit under is dense. If, in addition, the set of periodic points for is dense, then is said to be chaotic. Analogous definitions can also be stated for a -semigroup of linear and continuous operators in .
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 -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 is hypercyclic, then is hypercyclic for every [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 is hypercyclic if and only if is hypercyclic for every with [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 -semigroup 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 -semigroup and its operators.
The authors give an example of a chaotic operator such that is not chaotic for certain with . Besides, they give an example of a chaotic -semigroup such that there exists such that is chaotic and is not chaotic. Even more, they construct a chaotic -semigroup that does not contain any chaotic operator. The constructions are given from a detailed study of the point spectrum of chaotic operators.