Genericity of Homeomorphisms with Full Mean Hausdorff Dimension
It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and...
- Autores:
-
Muentes Acevedo, Jeovanny de Jesus
- Tipo de recurso:
- Fecha de publicación:
- 2024
- Institución:
- Universidad Tecnológica de Bolívar
- Repositorio:
- Repositorio Institucional UTB
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.utb.edu.co:20.500.12585/12668
- Acceso en línea:
- https://hdl.handle.net/20.500.12585/12668
- Palabra clave:
- Mean dimension
Metric mean dimension
Mean Hausdorff dimension
Hausdorff dimension
Topological entropy
LEMB
- Rights
- restrictedAccess
- License
- http://purl.org/coar/access_right/c_16ec
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Genericity of Homeomorphisms with Full Mean Hausdorff Dimension |
| title |
Genericity of Homeomorphisms with Full Mean Hausdorff Dimension |
| spellingShingle |
Genericity of Homeomorphisms with Full Mean Hausdorff Dimension Mean dimension Metric mean dimension Mean Hausdorff dimension Hausdorff dimension Topological entropy LEMB |
| title_short |
Genericity of Homeomorphisms with Full Mean Hausdorff Dimension |
| title_full |
Genericity of Homeomorphisms with Full Mean Hausdorff Dimension |
| title_fullStr |
Genericity of Homeomorphisms with Full Mean Hausdorff Dimension |
| title_full_unstemmed |
Genericity of Homeomorphisms with Full Mean Hausdorff Dimension |
| title_sort |
Genericity of Homeomorphisms with Full Mean Hausdorff Dimension |
| dc.creator.fl_str_mv |
Muentes Acevedo, Jeovanny de Jesus |
| dc.contributor.author.none.fl_str_mv |
Muentes Acevedo, Jeovanny de Jesus |
| dc.subject.keywords.spa.fl_str_mv |
Mean dimension Metric mean dimension Mean Hausdorff dimension Hausdorff dimension Topological entropy |
| topic |
Mean dimension Metric mean dimension Mean Hausdorff dimension Hausdorff dimension Topological entropy LEMB |
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LEMB |
| description |
It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities. Let N be an n -dimensional compact Riemannian manifold, where n⩾2 , and α∈[0,n] . In this paper, we construct a homeomorphism ϕ:N→N with mean Hausdorff dimension equal to α . Furthermore, we prove that the set of homeomorphisms on N with both lower and upper mean Hausdorff dimensions equal to α is dense in Hom(N) . Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to n contains a residual subset of Hom(N). |
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2024 |
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2024-05-07T11:56:21Z |
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2024-05-07T11:56:21Z |
| dc.date.issued.none.fl_str_mv |
2024-04-18 |
| dc.date.submitted.none.fl_str_mv |
2024-05-06 |
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Acevedo, J.M. Genericity of Homeomorphisms with Full Mean Hausdorff Dimension. Regul. Chaot. Dyn. (2024). https://doi.org/10.1134/S1560354724510014 |
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https://hdl.handle.net/20.500.12585/12668 |
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10.1134/S1560354724510014 |
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Universidad Tecnológica de Bolívar |
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Repositorio Universidad Tecnológica de Bolívar |
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Acevedo, J.M. Genericity of Homeomorphisms with Full Mean Hausdorff Dimension. Regul. Chaot. Dyn. (2024). https://doi.org/10.1134/S1560354724510014 10.1134/S1560354724510014 Universidad Tecnológica de Bolívar Repositorio Universidad Tecnológica de Bolívar |
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eng |
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eng |
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Campus Tecnológico |
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Regular and Chaotic Dynamics |
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Universidad Tecnológica de Bolívar |
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Muentes Acevedo, Jeovanny de Jesusec5c0208-d53f-44d4-a347-fdb3d28db2abColombia2024-05-07T11:56:21Z2024-05-07T11:56:21Z2024-04-182024-05-06Acevedo, J.M. Genericity of Homeomorphisms with Full Mean Hausdorff Dimension. Regul. Chaot. Dyn. (2024). https://doi.org/10.1134/S1560354724510014https://hdl.handle.net/20.500.12585/1266810.1134/S1560354724510014Universidad Tecnológica de BolívarRepositorio Universidad Tecnológica de BolívarIt is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities. Let N be an n -dimensional compact Riemannian manifold, where n⩾2 , and α∈[0,n] . In this paper, we construct a homeomorphism ϕ:N→N with mean Hausdorff dimension equal to α . Furthermore, we prove that the set of homeomorphisms on N with both lower and upper mean Hausdorff dimensions equal to α is dense in Hom(N) . Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to n contains a residual subset of Hom(N).17application/pdfengRegular and Chaotic DynamicsGenericity of Homeomorphisms with Full Mean Hausdorff Dimensioninfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_2df8fbb1http://purl.org/coar/version/c_970fb48d4fbd8a85Mean dimensionMetric mean dimensionMean Hausdorff dimensionHausdorff dimensionTopological entropyLEMBinfo:eu-repo/semantics/restrictedAccesshttp://purl.org/coar/access_right/c_16ecCartagena de IndiasCampus TecnológicoInvestigadoresAcevedo, J. M., Genericity of Continuous Maps with Positive Metric Mean Dimension, Results Math., 2022, vol. 77, no. 1, Paper No. 2, 30 pp.Acevedo, J. M., Romaña, S., and Arias, R., Density of the Level Sets of the Metric Mean Dimension for Homeomorphisms, J. Dyn. Diff. Equat., 2024 (in press).Acevedo, J. M., Baraviera, A., Becker, A. J., and Scopel, É., Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric, Preprint (2022).Artin, M. and Mazur, B., On Periodic Points, Ann. of Math. (2), 1965, vol. 81, pp. 82–99.Backes, L. and Rodrigues, F. B., A Variational Principle for the Metric Mean Dimension of Level Sets, IEEE Trans. Inform. Theory, 2023, vol. 69, no. 9, pp. 5485–5496.Cheng, D., Li, Zh., and Selmi, B., Upper Metric Mean Dimensions with Potential on Subsets, Nonlinearity, 2021, vol. 34, no. 2, pp. 852–867.Dou, D., Minimal Subshifts of Arbitrary Mean Topological Dimension, Discrete Contin. Dyn. Syst., 2017, vol. 37, no. 3, pp. 1411–1424.Carvalho, M., Rodrigues, F. B., and Varandas, P., Generic Homeomorphisms Have Full Metric Mean Dimension, Ergodic Theory Dynam. Systems, 2022, vol. 42, no. 1, pp. 40–64.Falconer, K., Fractal Geometry: Mathematical Foundations and Applications, 3rd ed., Chichester: Wiley, 2014.Furstenberg, H., Disjointness in Ergodic Theory, Minimal Sets, and a Problem in Diophantine Approximation, Math. Systems Theory, 1967, vol. 1, no. 1, pp. 1–49.Gromov, M., Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: 1, Math. Phys. Anal. Geom., 1999, vol. 2, no. 4, pp. 323–415.Gutman, Y., Embedding Topological Dynamical Systems with Periodic Points in Cubical Shifts, Ergodic Theory Dynam. Systems, 2017, vol. 37, no. 2, pp. 512–538.Gutman, Y., Qiao, Y., and Tsukamoto, M., Application of Signal Analysis to the Embedding Problem of -Actions, Geom. Funct. Anal., 2019, vol. 29, no. 5, pp. 1440–1502.Hurley, M., On Proofs of the General Density Theorem, Proc. Amer. Math. Soc., 1996, vol. 124, no. 4, pp. 1305–1309.Lindenstrauss, E. and Tsukamoto, M., Double Variational Principle for Mean Dimension, Geom. Funct. Anal., 2019, vol. 29, no. 4, pp. 1048–1109.Lindenstrauss, E. and Tsukamoto, M., From Rate Distortion Theory to Metric Mean Dimension: Variational Principle, IEEE Trans. Inform. Theory, 2018, vol. 64, no. 5, pp. 3590–3609.Lindenstrauss, E. and Weiss, B., Mean Topological Dimension, Israel J. Math., 2000, vol. 115, no. 1, pp. 1–24.Lindenstrauss, E., Mean Dimension, Small Entropy Factors and an Embedding Theorem, Inst. Hautes Études Sci. Publ. Math., 1999, no. 89, pp. 227–262.Liu, Y., Selmi, B., and Li, Zh., On the Mean Fractal Dimensions of the Cartesian Product Sets, Chaos Solitons Fractals, 2024, vol. 180, Paper No. 114503, 9 pp.Ma, X., Yang, J., and Chen, E., Mean Topological Dimension for Random Bundle Transformations, Ergodic Theory Dynam. Systems, 2019, vol. 39, no. 4, pp. 1020–1041.Tsukamoto, M., Mean Dimension of Full Shifts, Israel J. Math., 2019, vol. 230, no. 1, pp. 183–193.Tsukamoto, M., Mean Hausdorff Dimension of Some Infinite Dimensional Fractals, https://arxiv.org/abs/2209.00512 (2022).Velozo, A. and Velozo, R., Rate Distortion Theory, Metric Mean Dimension and Measure Theoretic Entropy, https://arxiv.org/abs/1707.05762 (2017).Walters, P., An Introduction to Ergodic Theory, Grad. Texts in Math., vol. 79, Berlin: Springer, 2000.Yang, R., Chen, E., and Zhou, X., Bowen’s Equations for Upper Metric Mean Dimension with Potential, Nonlinearity, 2022, vol. 35, no. 9, pp. 4905–4938.Yano, K., A Remark on the Topological Entropy of Homeomorphisms, Invent. 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