2016/06/21 by John R. Burke, Burke, John R., Leonardo Pinheiro +1
Computer Science · Mathematics · #37B05 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1606.06764
openalex publication_date 2016/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish semigroup (with an identity) can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said semigroup. This shows that every countable infinite product of an abelian polish semigroup has a topologically mixing endomorphism and, in particular, the countable infinite toroidal group has infinitely many topologically mixing endomorphisms.