Lyapunov minimizing measures for expanding maps of the circle
Autor(a) principal: | |
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Data de Publicação: | 2001 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFRGS |
Texto Completo: | http://hdl.handle.net/10183/27431 |
Resumo: | We consider the set of maps f є Fα+=Uβ >αC1β of the circle which are covering maps of degree D, expanding, minxєS1 f ¹(x) > 1 and orientation preserving. We are interested in characterizing the set of suchmaps f which admit a unique f -invariant probability measure _ minimizing ∫1n f ¹ d µ over all f -invariant probability measures. We show there exists a set G+ C Fα+, open and dense in the C1+α topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C1+α topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy. We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8]. We will also present some results on the set of f -invariant measures µ maximizing ∫ A dµ for a fixed C1-expanding map f and a general potential A, not necessarily equal to −ln f ¹. |
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Contreras, GonzaloLopes, Artur OscarThieullen, Ph.2011-01-15T05:58:57Z20010143-3857http://hdl.handle.net/10183/27431000305892We consider the set of maps f є Fα+=Uβ >αC1β of the circle which are covering maps of degree D, expanding, minxєS1 f ¹(x) > 1 and orientation preserving. We are interested in characterizing the set of suchmaps f which admit a unique f -invariant probability measure _ minimizing ∫1n f ¹ d µ over all f -invariant probability measures. We show there exists a set G+ C Fα+, open and dense in the C1+α topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C1+α topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy. We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8]. We will also present some results on the set of f -invariant measures µ maximizing ∫ A dµ for a fixed C1-expanding map f and a general potential A, not necessarily equal to −ln f ¹.application/pdfengErgodic theory and dynamical systems. Cambridge. Vol. 21, no. 5 (2001), p. 1379-1409.Medidas minimizantesExpansões de funções no círculoMedidas de LyapunovLyapunov minimizing measures for expanding maps of the circleEstrangeiroinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFRGSinstname:Universidade Federal do Rio Grande do Sul (UFRGS)instacron:UFRGSORIGINAL000305892.pdf000305892.pdfTexto completo (inglês)application/pdf273181http://www.lume.ufrgs.br/bitstream/10183/27431/1/000305892.pdf79c43b1a282c82e65f6b4d63b96dffc2MD51TEXT000305892.pdf.txt000305892.pdf.txtExtracted Texttext/plain79984http://www.lume.ufrgs.br/bitstream/10183/27431/2/000305892.pdf.txt0b57c72ce8690ff49524b7b5fbc687daMD52THUMBNAIL000305892.pdf.jpg000305892.pdf.jpgGenerated Thumbnailimage/jpeg1405http://www.lume.ufrgs.br/bitstream/10183/27431/3/000305892.pdf.jpg423c41b7a3b54bad09ca638a980ee331MD5310183/274312021-06-26 04:41:47.583323oai:www.lume.ufrgs.br:10183/27431Repositório de PublicaçõesPUBhttps://lume.ufrgs.br/oai/requestopendoar:2021-06-26T07:41:47Repositório Institucional da UFRGS - Universidade Federal do Rio Grande do Sul (UFRGS)false |
dc.title.pt_BR.fl_str_mv |
Lyapunov minimizing measures for expanding maps of the circle |
title |
Lyapunov minimizing measures for expanding maps of the circle |
spellingShingle |
Lyapunov minimizing measures for expanding maps of the circle Contreras, Gonzalo Medidas minimizantes Expansões de funções no círculo Medidas de Lyapunov |
title_short |
Lyapunov minimizing measures for expanding maps of the circle |
title_full |
Lyapunov minimizing measures for expanding maps of the circle |
title_fullStr |
Lyapunov minimizing measures for expanding maps of the circle |
title_full_unstemmed |
Lyapunov minimizing measures for expanding maps of the circle |
title_sort |
Lyapunov minimizing measures for expanding maps of the circle |
author |
Contreras, Gonzalo |
author_facet |
Contreras, Gonzalo Lopes, Artur Oscar Thieullen, Ph. |
author_role |
author |
author2 |
Lopes, Artur Oscar Thieullen, Ph. |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Contreras, Gonzalo Lopes, Artur Oscar Thieullen, Ph. |
dc.subject.por.fl_str_mv |
Medidas minimizantes Expansões de funções no círculo Medidas de Lyapunov |
topic |
Medidas minimizantes Expansões de funções no círculo Medidas de Lyapunov |
description |
We consider the set of maps f є Fα+=Uβ >αC1β of the circle which are covering maps of degree D, expanding, minxєS1 f ¹(x) > 1 and orientation preserving. We are interested in characterizing the set of suchmaps f which admit a unique f -invariant probability measure _ minimizing ∫1n f ¹ d µ over all f -invariant probability measures. We show there exists a set G+ C Fα+, open and dense in the C1+α topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the C1+α topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy. We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous equation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8]. We will also present some results on the set of f -invariant measures µ maximizing ∫ A dµ for a fixed C1-expanding map f and a general potential A, not necessarily equal to −ln f ¹. |
publishDate |
2001 |
dc.date.issued.fl_str_mv |
2001 |
dc.date.accessioned.fl_str_mv |
2011-01-15T05:58:57Z |
dc.type.driver.fl_str_mv |
Estrangeiro info:eu-repo/semantics/article |
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info:eu-repo/semantics/publishedVersion |
format |
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publishedVersion |
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http://hdl.handle.net/10183/27431 |
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0143-3857 |
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000305892 |
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http://hdl.handle.net/10183/27431 |
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eng |
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eng |
dc.relation.ispartof.pt_BR.fl_str_mv |
Ergodic theory and dynamical systems. Cambridge. Vol. 21, no. 5 (2001), p. 1379-1409. |
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openAccess |
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