Transitive and series transitive maps on R < sup > d < /sup >

Abstract

Motivated by the behavior of topologically transitive homomorphisms of Polish abelian groups, we say a continuous map f:Rd→Rd is ‘series transitive’ if for any two nonempty open sets U,V⊂Rd, there exist x∈U and n∈N such that ∑j=0n−1fj(x)∈V. We show that any map on a discrete and closed subset of Rd can be extended to a mixing map of Rd, and use this result to produce a mixing map f:Rd→Rd (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 Rd. We also construct a chaotic map (i.e., a transitive map with a dense set of periodic points) f:Rd→Rd 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.