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Seminar on Hyperbolicity and Shadowing in Linear Dynamics
Dario Darji
Last update: Pyrih Pavel, doc. RNDr., CSc. (08.07.2024)
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An active participation on the seminar. Last update: Pyrih Pavel, doc. RNDr., CSc. (06.07.2024)
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Some Resources: • Pilyugin, Sergei Yu.; Sakai, Kazuhiro Shadowing and hyperbolicity. Lecture Notes in Mathematics, 2193. Springer, Cham, 2017. xiv+216 pp. ISBN: 978-3-319-65183-5; 978-3-319-65184-2. • Bernardes, Nilson C., Jr.; Cirilo, Patricia R.; Darji, Udayan B.; Messaoudi, Ali; Pujals, Enrique R. Expansivity and shadowing in linear dynamics. J. Math. Anal. Appl. 461 (2018), no. 1, 796-816. • Bernardes, Nilson C., Jr.; Messaoudi, Ali Shadowing and structural stability for operators. Ergodic Theory Dynam. Systems 41 (2021), no. 4, 961-980. • Cirilo, Patricia; Gollobit, Bryce; Pujals, Enrique Dynamics of generalized hyperbolic linear operators. Adv. Math. 387 (2021), Paper No. 107830, 37 pp. • D'Aniello, Emma; Darji, Udayan B.; Maiuriello, Martina Generalized hyperbolicity and shadowing in Lp spaces. J. Differential Equations 298 (2021), 68-94. • Bayart, Frédéric Two problems on weighted shifts in linear dynamics. Proc. Amer. Math. Soc. 149 (2021), no. 12, 5255-5266. • D'Aniello, Emma; Darji, Udayan B.; Maiuriello, Martina Shift-like operators on Lp(X). J. Math. Anal. Appl. 515 (2022), no. 1, Paper No. 126393, 13 pp • Lopes, Artur O.; Vargas, Victor Entropy, pressure, ground states and calibrated sub-actions for linear dynamics. Bull. Braz. Math. Soc. (N.S.) 53 (2022), no. 3, 1073-1106. • Bernardes, Nilson C., Jr.; Messaoudi, Ali A generalized Grobman-Hartman theorem. Proc. Amer. Math. Soc. 148 (2020), no. 10, 4351-4360. • Darji, Udayan B.; Gonçalves, Daniel; Sobottka, Marcelo Shadowing, finite order shifts and ultrametric spaces. Adv. Math. 385 (2021), Paper No. 107760, 34 pp • Lee, K.; Morales, C. A. Hyperbolicity, shadowing, and bounded orbits. Qual. Theory Dyn. Syst. 21 (2022), no. 3, Paper No. 61, 13 pp. Last update: Pyrih Pavel, doc. RNDr., CSc. (08.07.2024)
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Description: In the 1960’s and 1970’s series of interesting articles, Hedlund and Eisenberg investigated relationship between various types of expansivity and the spectrum of linear operators on Banach spaces. During the last ten years, there has been a resurgence of interest in hyperbolic dynamics of linear operators on Banach spaces. This renewed interest began with the 2018 article of Bernardes-Cirilo-Darji-Messaoudi-Pujals. In this article, relationships between expansivity, shadowing and hyperbolicity in the setting on infinite dimensional Banach spaces were investigated. Whereas hyperbolic dynamics of finite dimensional Banach space is well-understood and simple, many interesting an open problems appear in the setting of infinite dimensional setting. In this seminar, we will discuss recent development, open questions and possible future directions.
Following is a tentative schedule for the course.
• 2 Weeks. A quick overview of hyperbolic dynamics on manifold. In particular, notions of expansivity, the shadowing property, hyperbolicity, and structural stability.
• 4 Weeks. In finite dimensional Banach spaces, the notions of expansivity, shadowing, hyperbolicity, and structural stability coincide. In infinite dimensional Banach spaces, hyperbolicity implies each of expansivity, the shadowing property and structural stability. However, examples showing that the reverse does not hold. We will carefully go through classical results of Hedlund and Eisenberg as well as the recent 2018 article of Bernardes-Cirilo-Darji-Messaoudi-Pujals.
• 5 Weeks. Generalized Hyperbolicity and its relation to various types of shadowing properties. We will carefully go through 2020/2021 articles of Bernardes and Messaoudi, as well as 2021 article of Cirilo, Gollobit and Pujals. The question of whether the shadowing property implies generalized hyperbolicity remains open.
• 4 weeks. Relationships between various types of shadowing properties. We will discuss some results which show that in certain special situations, an operator on Banach with the shadowing property does imply that the operator is generalized hyperbolic. This will be encompassed in discussion of 2020/2021 articles of D’Aniello-Darji-Maiuriello.
• 4 weeks. Relationship between structural stability, the shadowing property and generalized hyperbolicity. We will discuss 2021 article of Bayart which showed that for weighted backward shifts, structural stability implies that it has the shadowing property. We will ponder on what happens the general setting.
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 hyperbolic linear dynamics 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.
Last update: Pyrih Pavel, doc. RNDr., CSc. (08.07.2024)
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Prerequisites: Basics of real analysis, functional analysis and general topology. Last update: Pyrih Pavel, doc. RNDr., CSc. (06.07.2024)
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