On the Existence of Polynomials with Chaotic Behaviour
We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic)...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2013/320961 |
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| author | Nilson C. Bernardes Alfredo Peris |
| author_facet | Nilson C. Bernardes Alfredo Peris |
| author_sort | Nilson C. Bernardes |
| collection | DOAJ |
| description | We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold. |
| format | Article |
| id | doaj-art-9927f9d259244e73a3e2ce2a690bdc94 |
| institution | Kabale University |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-9927f9d259244e73a3e2ce2a690bdc942025-02-03T01:11:07ZengWileyJournal of Function Spaces and Applications0972-68021758-49652013-01-01201310.1155/2013/320961320961On the Existence of Polynomials with Chaotic BehaviourNilson C. Bernardes0Alfredo Peris1Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68530, 21945-970 Rio de Janeiro, RJ, BrazilIUMPA, Universitat Politècnica de València, Departament de Matemática Aplicada, Edifici 7A, 46022 València, SpainWe establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.http://dx.doi.org/10.1155/2013/320961 |
| spellingShingle | Nilson C. Bernardes Alfredo Peris On the Existence of Polynomials with Chaotic Behaviour Journal of Function Spaces and Applications |
| title | On the Existence of Polynomials with Chaotic Behaviour |
| title_full | On the Existence of Polynomials with Chaotic Behaviour |
| title_fullStr | On the Existence of Polynomials with Chaotic Behaviour |
| title_full_unstemmed | On the Existence of Polynomials with Chaotic Behaviour |
| title_short | On the Existence of Polynomials with Chaotic Behaviour |
| title_sort | on the existence of polynomials with chaotic behaviour |
| url | http://dx.doi.org/10.1155/2013/320961 |
| work_keys_str_mv | AT nilsoncbernardes ontheexistenceofpolynomialswithchaoticbehaviour AT alfredoperis ontheexistenceofpolynomialswithchaoticbehaviour |