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Tuesday Seminar on Topology
Seminar information archive ～01/05｜Next seminar｜Future seminars 01/06～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | HABIRO Kazuo, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
2026/01/13
17:00-18:00 Room #hybrid/056 (Graduate School of Math. Sci. Bldg.)
Pre-registration required. See our seminar webpage.
Sogo Murakami (The University of Tokyo)
On the shadowing property of differentiable dynamical systems beyond structural stability (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Pre-registration required. See our seminar webpage.
Sogo Murakami (The University of Tokyo)
On the shadowing property of differentiable dynamical systems beyond structural stability (JAPANESE)
[ Abstract ]
The shadowing property, which has been extensively studied in connection with hyperbolic differentiable dynamical systems, is a dynamical concept ensuring that approximate orbits with small errors (commonly referred to as pseudo-orbits) can be traced by a true orbit. This property is one of the fundamental notions closely related to structural stability. In this talk, I will present the conditions under which the shadowing property holds for differentiable dynamical systems that are not structurally stable, in both discrete-time and continuous-time settings. In the first part of the talk, conditions guaranteeing the shadowing property for Axiom A diffeomorphisms will be discussed. In particular, I will explain the T^{s,u}-condition, and its relationship with the C^0-transversality condition introduced by PetrovPilyugin. I will then give a sufficient condition for having the shadowing property for Axiom A diffeomorphisms. In the second part, results concerning the shadowing property on chain recurrent sets for flows will be presented. While Robinson (1977) showed that every hyperbolic set exhibits the shadowing property, it is known that no singular hyperbolic set with non-isolated hyperbolic singularity, such as the Lorenz attractor, admits the shadowing property (Wen-Wen, 2020). Motivated by this, Arbieto et al. conjectured that any chain recurrent set with attached (non-isolated) hyperbolic singularities cannot possess the shadowing property. In this talk, a counterexample to this conjecture will be constructed.
[ Reference URL ]The shadowing property, which has been extensively studied in connection with hyperbolic differentiable dynamical systems, is a dynamical concept ensuring that approximate orbits with small errors (commonly referred to as pseudo-orbits) can be traced by a true orbit. This property is one of the fundamental notions closely related to structural stability. In this talk, I will present the conditions under which the shadowing property holds for differentiable dynamical systems that are not structurally stable, in both discrete-time and continuous-time settings. In the first part of the talk, conditions guaranteeing the shadowing property for Axiom A diffeomorphisms will be discussed. In particular, I will explain the T^{s,u}-condition, and its relationship with the C^0-transversality condition introduced by PetrovPilyugin. I will then give a sufficient condition for having the shadowing property for Axiom A diffeomorphisms. In the second part, results concerning the shadowing property on chain recurrent sets for flows will be presented. While Robinson (1977) showed that every hyperbolic set exhibits the shadowing property, it is known that no singular hyperbolic set with non-isolated hyperbolic singularity, such as the Lorenz attractor, admits the shadowing property (Wen-Wen, 2020). Motivated by this, Arbieto et al. conjectured that any chain recurrent set with attached (non-isolated) hyperbolic singularities cannot possess the shadowing property. In this talk, a counterexample to this conjecture will be constructed.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
