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Seminář o chaosu a lineární dynamice
Dario Darji
Poslední úprava: Pyrih Pavel, doc. RNDr., CSc. (08.07.2024)
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Podmínkou pro zápočet je aktivní účast na semináři. Poslední úprava: Pyrih Pavel, doc. RNDr., CSc. (06.07.2024)
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Textbook/Resources:
· Grosse-Erdmann, Karl-G.; Peris Manguillot, Alfredo Linear chaos. Universitext. Springer, London, 2011. xii+386 pp. ISBN: 978-1-4471-2169-5
· Bayart, Frédéric; Matheron, Étienne Dynamics of linear operators. Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge, 2009. xiv+337 pp. ISBN: 978-0-521-51496-5
· Grivaux, S.; Matheron, É.; Menet, Q. Linear dynamical systems on Hilbert spaces: typical properties and explicit examples. Mem. Amer. Math. Soc. 269 (2021), no. 1315, v+147 pp. ISBN: 978-1-4704-4663-5; 978-1-4704-6468-4
· Cutting edge research articles mentioned in Reference List. Poslední úprava: Pyrih Pavel, doc. RNDr., CSc. (08.07.2024)
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Description:
This will be an introductory course on Chaos in Linear Dynamics quickly leading to current research topics. We will begin by discussing basic concepts of topological dynamics such as transitivity, mixing, Li-Yorke chaos, Devaney chaos, topological entropy and invariant measures. Then, we will discuss how these notions arise in the setting of linear dynamics in Banach spaces. By the end of the semester, the participants will be able to understand and appreciate open problems and current research in the area.
Following is a tentative weekly schedule for the course.
· Dynamical systems, Hypercyclicity, Mixing, Chaos, Li-Yorke Chaos.
· Frequently Hypercyclicity, Invariant measures, Topological Entropy
· Three Examples: Differentiation Operator, Translation Operator, Rolewicz Operator,
· Hypercyclic Criteria, Frequently Hypercyclic Criteria.
· Weighted Backward Shifts: Transitivity, Mixing, Li-Yorke Chaos.
· Weighted Backward Shifts: Frequent Hypercyclicity, Devaney Chaos, Bayart-Rusza Theorem.
· Composition Operators: Transitivity, Mixing, Li-Yorke chaos. Results of Bernardes-Darji-Pires.
· Composition Operator: Chaos and Frequently Hypercyclicity. Results of Darji-Pires. Open problems concerning relationship between chaotic operators and frequently hypercyclic operators.
· Odometers and Chaos. Results of Darji-D’Aniello-Bongiorno-DiPiazza. Open problems and ideas concerning how to distinguish chaotic, mixing and frequently hypercyclic operators. Result of Menet.
· Hopf Decomposition and Composition Operators on Conservative Systems.
· Ansari’s Theorem
· Frequently Hypercyclic Criteria, Bayart-Grivaux, Grivaux-Matheron Theorems and open problems concerning invariant measures and frequent hypercyclicity.
· A-hypercyclicity, Recurrent Operators. Recent results and current research in the areas.
Format: This will be a very interactive seminar with many discussions and presentations by participants. I will introduce topics by lecturing and leave the audience with many questions and ideas to ponder for the subsequent meeting. During the subsequent meeting, I will solicit participation from the audience. We will discuss their insights, thoughts, questions, doubts and novel ideas. We will also have assigned periodic presentations by some of the participants.
This is how I normally teach my classes at University of Louisville. My philosophy is to do is to learn.
The topic of the seminar will be background material for linear chaos problems of my research proposal. The material will be covered in a fashion so that interested participants can collaborate with me on my project in a short time. While the audience will become familiar classical results of the area, the emphasis will be on solving open problems and exploring new directions.
Poslední úprava: Pyrih Pavel, doc. RNDr., CSc. (08.07.2024)
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Prerequisites: Basics of real analysis and general topology. Poslední úprava: Vejnar Benjamin, doc. Mgr., Ph.D. (19.09.2024)
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