LAUSR

Abstract Motivated by the behavior of topologically transitive homomorphisms of Polish abelian groups, we say a continuous map f : R d → R d is ‘series transitive’ if for any two nonempty open sets U , V ⊂ R d , there exist x ∈ U and n ∈ N such that ∑ j = 0 n − 1 f j ( x ) ∈ V . We show that any map on a discrete and closed subset of R d can be extended to a mixing map of R d , and use this result to produce a mixing map f : R d → R d (for each d ∈ N ) which is also series transitive. We have examples to say that transitivity and series transitivity are independent properties for continuous self-maps of R d . We also construct a chaotic map (i.e., a transitive map with a dense set of periodic points) f : R d → R d such that f is arbitrarily close to and asymptotic to the identity map. Finally, we make a few observations about topological transitivity of continuous homomorphisms of Polish abelian groups.