Topological Transitivity of Algebraically Recurrent Sets
DOI:
https://doi.org/10.5644/SJM.20.02.12Keywords:
trajectory, algebraic recurrence, group, topological group, topological transitivity, non-wandering set, chain recurrent setAbstract
In this paper, we will discuss the connection between topological transitivity and recurrence of \( G \)-flows acting on a compact metric space \( X \). We will prove that the \( T T \)-property of the set of all algebraically recurrent points \( AR(\varphi) \) implies chain recurrent properties of the whole space and hence improve some of the results from [6].
Downloads
References
[1] J. Ayala, P. Corbin, K. McConville, F. Colonius, W. Kliemann, J. Peters: Morse decomposition, attractors and chain recurrence, Proyecciones Journal of Mathematics, Vol. 25, 2006, pp. 79-109.
[2] N. P. Bhatia, G. P. Szegö: Stability Theory of Dynamical Systems, Grundlehren der Math. Wiss., Vol. 161, Springer, Berlin, 1970.
[3] H. Fustenberg: Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton, New Jersey, 1981.
[4] T. Fisher, B. Hasselblatt: Hyperbolic flows, Graduate School of Mathematical Sciences, The University of Tokyo, 2018.
[5] E. Glasner: Classifying Dynamical Systems By Their Recurrence Properties, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center, Vol. 24, 2004, pp. 21-40.
[6] M. Mazur: Topological transitivity of the chain recurrent set implies topological transitivity of the whole space, Universitatis Iagellonicae Acta Mathematica, 2000, pp. 219-226.
[7] C. P. Niculescu: Topological transitivity and recurrence as a source of chaos, Functional Analysis and Economic Theory, 1998, pp. 101-108.
