Hiperciclicidade e caos linear

Abstract

In the last years, the linear dinamics has gained the attention of many researchers, mainly to the investigation of linear and continuous operators T : X −→ X, on topological vector spaces, whose orbit {x,Tx,...,Tnx,...} is dense for some x ∈ X. Operators with this behaviour are said to be hypercyclic and the theory that studies them is known by hypercyclicity, which is one of the main themes within this work. The three classical examples of hyperciclic operators found in the literature are investigated: the Birkhoff (1884−1944), MacLane (1909−2005) and Rolewicz (1932−2015) operators. The Devaney chaos, which has as one of its “ingredients” the phenomeon of hypercyclicity, is presented and the verification that the classic operators are Devaney chaotic is fulfilled. Among interesting results about hypercyclicity, are discussed somes criterions, the constatation that there are no hypercyclic operators on a finite dimensional space and a curious result: any hypercyclic operator admits a dense invariant subspace consisting, except for zero, of hypercyclic vectors. Ultimately, a brief discussion is presented about another two types of chaos, namely the Li-Yorke and distributional chaos.