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Shadowing of pseudotrajectories in dynamical systems
Tuesday, Thursday, 17:00 – 19:00,
PUC-Rio, Departamento de Matemática, 8 andar
The first lecture: 15 Aug
Whatsup group:
https://chat.whatsapp.com/CZjXtPJycOY4Ovxz8273yH
15.08 Lecture 1
22.08 Lecture 2
12.09 Lecture 3, 4 (17:00 -- 20:00)
17.09 Lecture 5 (17:00 -- 19:00)
24.09 Lecture 6, 7 (17:00 -- 20:00)
03.10 Lecture 8, 9 (17:00 -- 20:00)
10.10 Lecture 10 (18:00 -- 20:00)
24.10 Lecture 11, 12 (17:00 -- 20:00)
31.10 Lecture 13, 14 (17:00 -- 20:00)
7.11 Lecture 15, 16 (17:00 -- 20:00)
14.11 Lecture 17, 18 (17:00 -- 20:00)
21.11 Lecture 19, 20 (17:00 -- 20:00)
28.11 Lecture 21 (17:00 -- 20:00)
05.12 Exam. Questions.
Shadowing theory studies a question under which conditions properties of approximate trajectories (pseudotrajectories) are similar to properties of exact trajectories. This notion plays an important role in hyperbolic dynamical systems. It starts with well-known Anosov/Bowen shadowing lemma stating that in a neighborhood of a hyperbolic set dynamical system has the shadowing property. In the course we will cover different aspects: quantitative properties, differences between discrete dynamical systems and flows, group actions, probability aspects. The course is nicely couples with Hyperbolic Dynamics.
Approximate list of topics:
Revisiting of hyperbolicity, Exponential dichotomy, C-lambda structure,.
Shadowing lemma via contraction mapping (Anosov)
Shadowing lemma via geometric approach (Bowen)
Shadowing lemma via inhomogeneous linear equation.
Structural stability via shadowing and expansivity.
Lipschitz and Holder shadowing.
Set of diffeomorphisms with shadowing in $C^1$-topology.
Stochastic shadowing.
Inverse shadowing
Shadowing for flows. Role of reparametrization.
Examples of shadowing in a neighborhood of nontransverse heteroclinic.
Shadowing for group actions.
Prerequisits: Dynamica Hyperbolica, Master and PhD students.
References:
1. Pilyugin, Sergei Yu. Shadowing in dynamical systems. Lecture Notes in Math., 1706 Springer-Verlag, Berlin, 1999.
2. Palmer, Ken. Shadowing in dynamical systems. Theory and applications Math. Appl., 501 Kluwer Academic Publishers, Dordrecht, 2000.
3. Pilyugin, Sergei Yu.; Sakai, Kazuhiro, Shadowing and hyperbolicity. Lecture Notes in Math., 2193 Springer, Cham, 2017.
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