| Many examples of mathematical processes, each of which is divergent in some unpredictable and chaotic way, have been found and studied in history. We are interested in the study and search for what are often large vector spaces of functions on R or C which have special properties. Given such a property, we say that the subset M of functions on [0,1] which satisfies it is lineable (resp. spaceable) if M ∪ {0} contains an infinite dimensional subspace (resp. closed subspace). Moreover, we say that the set M is algebrable if M ∪ {0} contains an infinitely generated algebra. We are interested in the study of the lineability, spaceability and algebrability of sets of functions on R or C which are everywhere surjective, differentiable and nowhere monotone, non-measurable, or enjoying any of these so called special properties. We are also interested in the study of hypercyclic operators on Banach spaces. If X denotes a separable infinite dimensional Banach space and T : X → X a bounded linear operator on X, then we say that x ∈ X is a hypercyclic vector for T if its orbit, {Tnx : n ∈ N }, is dense in X. If there exists such an x ∈ X we call T a hypercyclic operator. We study the structure of the set of hypercyclic vectors for a given operator T, HC( T), and the lineability/algebrability of HC( T). We also give, and study, necessary and sufficient conditions for an operator to be chaotic or hypercyclic. |
