The main goal of this paper is the investigation of a relevant property which appears in the various definition of deterministic topological chaos for discrete time dynamical system: transitivity. Starting from the standard Devaney's notion of topological chaos based on regularity, transitivity, and sensitivity to the initial conditions, the critique formulated by Knudsen is taken into account in order to exclude periodic chaos from this definition. Transitivity (or some stronger versions of it) turns out to be the relevant condition of chaos and its role is discussed by a survey of some important results about it with the presentation of some new results. In particular, we study topological mixing, strong transitivity, and full transitivity. Their applications to symbolic dynamics are investigated with respect to the relationships with the associated languages.

New compact representations of infinite graphs are investigated. Finite automata are used to represent labelled hyper-graphs which can be also multi-graphs. Our approach consists of a general framework where vertices are represented by a regular prefix-free language and edges are represented by a regular language and a function over tuples. We consider three different functions over tuples: given a tuple the first function returns its first difference, the second one returns its suffix and the last one returns its infixes. The first-difference function is substantially a direct generalization to infinite multi-hyper-graphs of the representation introduced by Ehrenfeucht et al. for finite graphs. This representation, though very interesting for finite graphs, turns out to be quite unsatisfactory for infinite graphs. The other two functions we consider while preserving some interesting features of their representation also achieves a high expressive power. As a matter of fact, our formalism either with the suffix or infix function results to be more powerful than the equational graphs introduced by Courcelle and the simple graphs defined by Caucal. The monadic second order theories of these two classes of graphs are undecidable, but still many interesting graph properties are decidable. The use of a regular prefix-free language to represent the vertices allows (fixed the language of the edges) to express a graph by a labelled tree, moreover, the use of finite automata to represent the edges allows the verification of graph properties.