Genericity of Continuous Maps with Positive Metric Mean Dimension
M. Gromov introduced the mean dimension for a continuous map in the late 1990’s, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy...
- Autores:
-
Muentes Acevedo, Jeovanny
- Tipo de recurso:
- Fecha de publicación:
- 2021
- Institución:
- Universidad Tecnológica de Bolívar
- Repositorio:
- Repositorio Institucional UTB
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.utb.edu.co:20.500.12585/10441
- Acceso en línea:
- https://hdl.handle.net/20.500.12585/10441
https://doi.org/10.1007/s00025-021-01513-3
- Palabra clave:
- Mean dimension
Metric mean dimension
Topological entropy
Box dimension
Hausdorff dimension
LEMB
- Rights
- openAccess
- License
- http://creativecommons.org/licenses/by-nc-nd/4.0/
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| dc.title.spa.fl_str_mv |
Genericity of Continuous Maps with Positive Metric Mean Dimension |
| title |
Genericity of Continuous Maps with Positive Metric Mean Dimension |
| spellingShingle |
Genericity of Continuous Maps with Positive Metric Mean Dimension Mean dimension Metric mean dimension Topological entropy Box dimension Hausdorff dimension LEMB |
| title_short |
Genericity of Continuous Maps with Positive Metric Mean Dimension |
| title_full |
Genericity of Continuous Maps with Positive Metric Mean Dimension |
| title_fullStr |
Genericity of Continuous Maps with Positive Metric Mean Dimension |
| title_full_unstemmed |
Genericity of Continuous Maps with Positive Metric Mean Dimension |
| title_sort |
Genericity of Continuous Maps with Positive Metric Mean Dimension |
| dc.creator.fl_str_mv |
Muentes Acevedo, Jeovanny |
| dc.contributor.author.none.fl_str_mv |
Muentes Acevedo, Jeovanny |
| dc.subject.keywords.spa.fl_str_mv |
Mean dimension Metric mean dimension Topological entropy Box dimension Hausdorff dimension |
| topic |
Mean dimension Metric mean dimension Topological entropy Box dimension Hausdorff dimension LEMB |
| dc.subject.armarc.none.fl_str_mv |
LEMB |
| description |
M. Gromov introduced the mean dimension for a continuous map in the late 1990’s, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if N is an n dimensional compact riemannian manifold then, for any a ∈ [0, n], the set consisting of continuous maps with metric mean dimension equal to a is dense in C0(N) and for a = n this set is residual. Furthermore, we prove some results related to the existence and, density of continuous maps, defined on Cantor sets, with positive metric mean dimension and also continous maps, defined on product spaces, with positive mean dimension. |
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2021 |
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2021-11-02 |
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2022-02-08T12:30:31Z |
| dc.date.available.none.fl_str_mv |
2022-02-08T12:30:31Z |
| dc.date.submitted.none.fl_str_mv |
2022-02-04 |
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http://purl.org/coar/resource_type/c_2df8fbb1 |
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Acevedo, Jeovanny De Jesus. (2022). Genericity of Continuous Maps with Positive Metric Mean Dimension. Results in Mathematics. 77. 10.1007/s00025-021-01513-3. |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/20.500.12585/10441 |
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https://doi.org/10.1007/s00025-021-01513-3 |
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Universidad Tecnológica de Bolívar |
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Repositorio Universidad Tecnológica de Bolívar |
| identifier_str_mv |
Acevedo, Jeovanny De Jesus. (2022). Genericity of Continuous Maps with Positive Metric Mean Dimension. Results in Mathematics. 77. 10.1007/s00025-021-01513-3. Universidad Tecnológica de Bolívar Repositorio Universidad Tecnológica de Bolívar |
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https://hdl.handle.net/20.500.12585/10441 https://doi.org/10.1007/s00025-021-01513-3 |
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eng |
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eng |
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Attribution-NonCommercial-NoDerivatives 4.0 Internacional |
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openAccess |
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30 Páginas |
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application/pdf |
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Colombia |
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Cartagena de Indias |
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Results in Mathematics, vol. 77, N° 2 (2022) |
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Universidad Tecnológica de Bolívar |
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Muentes Acevedo, Jeovannya00a2fcf-ed20-430d-a46b-2195dd9675d5Colombia2022-02-08T12:30:31Z2022-02-08T12:30:31Z2021-11-022022-02-04Acevedo, Jeovanny De Jesus. (2022). Genericity of Continuous Maps with Positive Metric Mean Dimension. Results in Mathematics. 77. 10.1007/s00025-021-01513-3.https://hdl.handle.net/20.500.12585/10441https://doi.org/10.1007/s00025-021-01513-3Universidad Tecnológica de BolívarRepositorio Universidad Tecnológica de BolívarM. Gromov introduced the mean dimension for a continuous map in the late 1990’s, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if N is an n dimensional compact riemannian manifold then, for any a ∈ [0, n], the set consisting of continuous maps with metric mean dimension equal to a is dense in C0(N) and for a = n this set is residual. Furthermore, we prove some results related to the existence and, density of continuous maps, defined on Cantor sets, with positive metric mean dimension and also continous maps, defined on product spaces, with positive mean dimension.30 Páginasapplication/pdfenghttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://purl.org/coar/access_right/c_abf2Results in Mathematics, vol. 77, N° 2 (2022)Genericity of Continuous Maps with Positive Metric Mean Dimensioninfo:eu-repo/semantics/articleinfo:eu-repo/semantics/restrictedAccesshttp://purl.org/coar/resource_type/c_2df8fbb1Mean dimensionMetric mean dimensionTopological entropyBox dimensionHausdorff dimensionLEMBCartagena de IndiasArtin, M., Mazur, B.: On periodic points. Ann. Math., 82–99 (1965)Block, L.: Noncontinuity of topological entropy of maps of the Cantor set and of the interval. Proc. Am. Math. Soc. 50(1), 388–393 (1975)Bobok, J., Zindulka, O.: Topological entropy on zero-dimensional spaces. Fundam. Math. 162(3), 233–249 (1999)Carvalho, M., Rodrigues, F.B., Varandas, P.: Generic homeomorphisms have full metric mean dimension. Ergodic Theory Dyn. Syst., 1–25 (2019)De Melo, W., Van Sebastian, S.: One-Dimensional Dynamics, vol. 25. Springer, Berlin (2012)do Carmo., M.P.: Geometria riemanniana. Instituto de Matem´atica Pura e Aplicada (2008)Engelking, R.: “General Topology. Heldermann, Berlin.” MR1039321 (91c: 54001): 529 (1989)Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Hoboken (2004)Gromov, M.: Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2(4), 323–415 (1999)Gutman, Y.: Embedding topological dynamical systems with periodic points in cubical shifts. Ergodic Theory Dyn. Syst. 37(2), 512–538 (2017)Gutman, Y., Tsukamoto, M.: Embedding minimal dynamical systems into Hilbert cubes. Invent. Math. 221(1), 113–166 (2020)Hurley, M.: On proofs of the C0 general density theorem. Proc. Am. Math. Soc. 124(4), 1305–1309 (1996)Jin, L., Yixiao, Q.: Mean dimension of product spaces: a fundamental formula. arXiv preprint arXiv:2102.10358 (2021)Lindenstrauss, E.: Mean dimension, small entropy factors and an embedding theorem. Publications Math´ematiques de l‘Institut des Hautes Etudes Scientifiques ´ 89(1), 227–262 (1999)] Lindenstrauss, E., Weiss, B.: Mean topological dimension. Israel J. Math. 115(1), 1–24 (2000)Lindenstrauss, E., Tsukamoto, M.: From rate distortion theory to metric mean dimension: variational principle. IEEE Trans. Inf. Theory 64(5), 3590–3609 (2018)Lindenstrauss, E., Tsukamoto, M.: Mean dimension and an embedding problem: an example. Israel J. Math. 199(2), 573–584 (2014)Misiurewicz, M.: Horseshoes for Continuous Mappings of an Interval. Dynamical Systems, pp. 125–135. Springer, Berlin (2010)Newhouse, S.E.: Continuity properties of entropy. Ann. Math. 129(1), 215–235 (1989)Rodrigues, F.B., Jeovanny, M.A.: Mean dimension and metric mean dimension for non-autonomous dynamical systems. J. Dyn. Control Syst. 1–27 (2021)Tsukamoto, M.: Mean dimension of full shifts. Israel J. Math. 230(1), 183–193 (2019)Velozo, A., Renato, V.: Rate distortion theory, metric mean dimension and measure theoretic entropy. arXiv preprint arXiv:1707.05762 (2017)Wei, C., Wen, S., Wen, Z.: Remarks on dimensions of Cartesian product sets. Fractals 24(03), 1650031 (2016)Yano, K.: A remark on the topological entropy of homeomorphisms. Invent. 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