Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces Article Swipe
Nilson C. Bernardes
,
Blas M. Caraballo
,
Udayan B. Darji
,
Vinícius V. Fávaro
,
Alfredo Peris
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2403.02843
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2403.02843
We introduce and study the notions of (generalized) hyperbolicity, topological stability and (uniform) topological expansivity for operators on locally convex spaces. We prove that every generalized hyperbolic operator on a locally convex space has the finite shadowing property. Contrary to what happens in the Banach space setting, hyperbolic operators on Fréchet spaces may fail to have the shadowing property, but we find additional conditions that ensure the validity of the shadowing property. Assuming that the space is sequentially complete, we prove that generalized hyperbolicity implies the strict periodic shadowing property, but we also show that the hypothesis of sequential completeness is essential. We show that operators with the periodic shadowing property on topological vector spaces have other interesting dynamical behaviors, including the fact that the restriction of such an operator to its chain recurrent set is topologically mixing and Devaney chaotic. We prove that topologically stable operators on locally convex spaces have the finite shadowing property and the strict periodic shadowing property. As a consequence, topologically stable operators on Banach spaces have the shadowing property. Moreover, we prove that generalized hyperbolicity implies topological stability for operators on Banach spaces. We prove that uniformly topologically expansive operators on locally convex spaces are neither Li-Yorke chaotic nor topologically transitive. Finally, we characterize the notion of topological expansivity for weighted shifts on Fréchet sequence spaces. Several examples are provided.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2403.02843
- https://arxiv.org/pdf/2403.02843
- OA Status
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- Related Works
- 10
- OpenAlex ID
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https://openalex.org/W4392538317Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2403.02843Digital Object Identifier
- Title
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Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex SpacesWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-03-05Full publication date if available
- Authors
-
Nilson C. Bernardes, Blas M. Caraballo, Udayan B. Darji, Vinícius V. Fávaro, Alfredo PerisList of authors in order
- Landing page
-
https://arxiv.org/abs/2403.02843Publisher landing page
- PDF URL
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https://arxiv.org/pdf/2403.02843Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2403.02843Direct OA link when available
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Regular polygon, Stability (learning theory), Mathematics, Pure mathematics, Mathematical analysis, Computer science, Geometry, Machine learningTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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10Other works algorithmically related by OpenAlex
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