Topologically mixing extensions of endomorphisms on Polish groups




weak mixing, Polish group, hypercyclicity criterion


In this paper we study the dynamics of continuous endomorphisms on Polish groups. We offer necessary and sufficient conditions for a continuous endomorphism on a Polish group to be weakly mixing. We prove that any continuous endomorphism of an abelian Polish group can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said group.


Download data is not yet available.

Author Biographies

John Burke, Rhode Island College

Department of Mathematical Sciences

Leonardo Pinheiro, Rhode Island College

Associate Professor, 

Department of Mathematical Sciences


J. Bès and A. Peris, Hereditarily hypercyclic operators, Journal of Functional Analysis 167 (1999), 94-112.

G. D. Birkhoff, Surface transformations and their dynamical applications, Acta. Math. 43 (1922), 1-119.

K. Chan, Universal meromorphic functions, Complex Variables, Theory and Applications 46 (2001), 307-314.

C. Chan and G. Turcu, Chaotic extensions of operators on Hilbert subspaces, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matemáticas 105 (2011), 415-421.

M. Gethner and J. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proceedings of the American Mathematical Society 100 (1987), 281-288.

C. Kitai, Invariant closed sets for linear operators, University of Toronto Thesis (1982).

S. Kolyada and L'. Snoha, Topological Transitivity - a survey, Grazer Math. Ber. 334 (1997), 3-35.

M. de la Rosa and C. Reed, Invariant closed sets for linear operators, Journal of Operator Theory 61 (2009), 369-380.

T. K. Subrahmonian Moothathu, Weak mixing and mixing of a single transformation of a topological (semi)group, Aequationes Mathematicae 78 (2009), 147-155.




How to Cite

J. Burke and L. Pinheiro, “Topologically mixing extensions of endomorphisms on Polish groups”, Appl. Gen. Topol., vol. 23, no. 1, pp. 179–187, Apr. 2022.



Regular Articles