Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778059 | Topology and its Applications | 2017 | 14 Pages |
Abstract
Motivated by the behavior of topologically transitive homomorphisms of Polish abelian groups, we say a continuous map f:RdâRd is 'series transitive' if for any two nonempty open sets U,VâRd, there exist xâU and nâN such that âj=0nâ1fj(x)âV. 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:RdâRd (for each dâN) 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:RdâRd 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Geometry and Topology
Authors
T.K. Subrahmonian Moothathu,