Topologically mixing extensions of endomorphisms on Polish groups
Submitted: 2021-02-28
|Accepted: 2022-01-08
|Published: 2022-04-01
Copyright (c) 2022 Applied General Topology
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Downloads
Keywords:
weak mixing, Polish group, hypercyclicity criterion
Supporting agencies:
Abstract:
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.
References:
J. Bès and A. Peris, Hereditarily hypercyclic operators, Journal of Functional Analysis 167 (1999), 94-112. https://doi.org/10.1006/jfan.1999.3437
G. D. Birkhoff, Surface transformations and their dynamical applications, Acta. Math. 43 (1922), 1-119. https://doi.org/10.1007/BF02401754
K. Chan, Universal meromorphic functions, Complex Variables, Theory and Applications 46 (2001), 307-314. https://doi.org/10.1080/17476930108815418
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. https://doi.org/10.1007/s13398-011-0029-3
M. Gethner and J. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proceedings of the American Mathematical Society 100 (1987), 281-288. https://doi.org/10.1090/S0002-9939-1987-0884467-4
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. https://doi.org/10.1007/s00010-009-2958-x
