Linear dynamics of semigroups generated by differential operators

Abstract

During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C 0 -semigroups of linear and continuous operators.We will review some of these notions and we will discuss basic properties of the dynamics of C 0 -semigroups. We will also study in detail the dynamics of the translation C 0 -semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C 0 -semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models. . On the one hand, it is connected with the still unsolved Invariant Subspace Problem on Hilbert spaces, which asks for the existence of an operator with no nontrivial closed invariant subset. The answer was positive in Banach spaces, as it was seen by Read [4] in`1. On the other hand, there is a number of different connections of linear dynamics with different areas such as algebra, topology, real and complex analysis, functional analysis, approximation theory, number theory, and probability.The advances in the area were first compiled by Grosse-Erdmann [5,6]. The monographs of Bayart and Matheron [7] and Grosse-Erdmann and Peris [8] represent a good source of the state of the art in the area. The study of the size and the algebraic structure of the set of vectors with wild behaviour, provides an interesting field for continuing with the quest of surprising examples of sets of pathological elements that, however, are preserved by the elementary algebraic operators. In this line, a selection of topics was recently revisited by Aron et al. in [9, Ch. 4], see also [10].Given a family fT i g i 2I of linear and continuous operators on an infinite-dimensional separable Banach space X , we say that it is universal if there exists some element x 2 X such that, its orbit fT i x W i 2 I g is dense in X .

Frequently Hypercyclic Semigroup Generated by Some Partial Differential Equations with Delay Operator

“…Chaotic phenomena are interesting and abundant topics in different areas and attract many mathematicians (see, e.g., [1,2,[7][8][9][10]). Alberto Conejero et al [7] introduced different kinds of chaotic operators, such as Devaney chaos, frequent hypercyclicity, and so on. In [7] the authors provide a lot of examples.…”

“…Alberto Conejero et al [7] introduced different kinds of chaotic operators, such as Devaney chaos, frequent hypercyclicity, and so on. In [7] the authors provide a lot of examples. Here, we are interesting in a tape of C 0 -semigroup so called frequently hypercyclic semigroup.…”