Abstract: Devaney has given a popular definition of chaos in discrete dynamical systems. It comprises the three conditions (i) topological transitivity, (ii) a dense set of periodic points and (iii) sensitive dependence on initial conditions. This paper shows that the first two conditions imply the third. This not only simplifies the definition of chaos but also turns it into a purely topological one since it does away with the need to include the metric property of sensitive dependence on initial conditions in the definition. As a consequence chaos as defined by Devaney is always preserved by topological conjugacy.
For a layperson's introduction to chaos theory and the ideas explored in this paper have a look at my pictorial introduction to chaos.
Abstract: Smale's horseshoe map is a standard example in the study of discrete dynamical systems. It occurs in a wide range of physical problems and has chaotic dynamics on an invariant set. We use the ternary representation of the middle-thirds Cantor set to give a simpler treatment of a special but typical case of Smale's horseshoe map.
Abstract: One may often decompose the domain of a topologically transitive map into finitely many regular closed pieces with nowhere dense overlap in such a way that these pieces map into one another in a periodic fashion. We call decompositions of this kind regular periodic decompositions and refer to the number of pieces as the length of the decomposition.
If f is topologically transitive but its n-th iterate is not, then f has a regular periodic decomposition of some length dividing n. Although a decomposition of a given length is unique, a map may have many decompositions of different lengths. The set of lengths of decompositions of a given map is an ideal in the lattice of natural numbers ordered by divisibility, which we call the decomposition ideal of f.
Every ideal in this lattice arises as a decomposition ideal of some map. Decomposition ideals of Cartesian products of transitive maps are discussed and used to develop various examples. Results are obtained concerning the implications of local connectedness for decompositions. We conclude with a comprehensive analysis of the possible decomposition ideals for maps on 1-manifolds.
Abstract: Topological transitivity, weak mixing and non-wandering are definitions used in topological dynamics to describe the ways in which open sets feed into each other under iteration. Using finite directed graphs, we generalize these definitions to obtain topological mapping properties. We consider the extent to which these mapping properties are logically distinct. As it turns out, there are only three distinct properties which entail ``interesting" dynamics. Two of these, transitivity and weak mixing, are already well known. The third does not appear in the literature, but turns out to be just weak mixing with a slight structural variation. The remaining properties comprise a countably infinite collection of properties entailing somewhat less interesting dynamics. This latter collection includes non-wandering.