Abstract
A shadowable point for a flow is a point where the shadowing lemma holds for pseudo-orbits passing through it. We prove that this concept satisfies the following properties: the set of shadowable points is invariant and a G δ set. A flow has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is shadowable. The chain recurrent points which are shadowable are exactly those that can be are approximated by periodic points when the flow is expansive. These results extends those presented in Morales (Dyn Syst. 2016;31(3):347–356). We study the relations between shadowable points of a homeomorphism and the shadowable points of its suspension flow. We characterize the set of forward shadowable points for transitive flows and chain transitive flows. We prove that the geometric Lorenz attractor does not have shadowable points. We show that in the presence of shadowable points chain transitive flows are transitive and that transitivity is a necessary condition for chain recurrent flows with shadowable points whenever the phase space is connected. Finally, as an application, these results we give concise proofs of some well known theorems establishing that flows with POTP admitting some kind of recurrence are minimal.
Similar content being viewed by others
References
Alongi J, Nelson G, Vol. 85. Recurrence and topology graduate studies in mathematics. Providence: American Mathematical Society; 2007.
Araújo V., Pacifico M, Vol. 53. Three-dimensional flows. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A series of modern surveys in mathematics. Heidelberg: Springer; 2010.
Bowen R, Walters P. Expansive one-parameter flows. J Diff Equ 1972;12: 180–93.
He L, Wang Z. Distal flows with pseudo orbit tracing property. Chinese Sci Bull 1995;40(19):1585–8.
Mai J. Pointwise recurrent dynamical systems with pseudo-orbit tracing property. Northeast Math J 1996;12(1):73–8.
Kato K. Pseudo-orbits and stabilities of flows. Mem Foc Sci Kochi Univ (Math) 1984;5:5–45.
Kato K. Pseudo-orbits and stabilities of flows. General Topology Appl 1979;10(1): 67–85.
Kawaguchi N. Quantitative shadowable points. Dyn Syst. 1–15; 2017. https://doi.org/10.1080/14689367.2017.1280664.
Komuro M. One-parameter flows with the pseudo-orbit tracing property. Monatsh Math 1984;98(3):219–53.
Komuro M. Lorenz attractors do not have the pseudo-orbit tracing property. J Math Soc Japan 1985;37(3):489–514.
Morales CA. Shadowable points. Dyn Syst 2016;31(3):347–56.
Moothathu TKS. Implications of pseudo-orbit tracing property for continuous maps on compacta. Topology Appl 2011;158(16):2232–9.
Palmer K. Shadowing in dynamical systems: theory and applications. Alphen aan den Rijn: Kluwer; 2000.
Pilyugin SY. Shadowing in dynamical systems. Lecture notes in mathematics. Berlin: Springer-Verlag; 1999, p. 1706.
Thomas RF. Stability properties of one-parameter flows. Proc London Math Soc (3) 1982;45(3):479–505.
Williams R. The structure of Lorenz attractors. Inst Hautes Études Sci Publ Math. 1979;50:73–99.
Villavicencio H. \(\mathcal {F}\)-Expansivity for Borel measures. J.Diff Equ 2016;261(10):5350–70.
Funding
This study was partially supported by CAPES from Brazil and FONDECYT from Peru (C.G. 217–2014).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aponte, J., Villavicencio, H. Shadowable Points for Flows. J Dyn Control Syst 24, 701–719 (2018). https://doi.org/10.1007/s10883-017-9381-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-017-9381-8