-
Previous Article
Local exact controllability to trajectories of the magneto-micropolar fluid equations
- EECT Home
- This Issue
-
Next Article
On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems
The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors
| 1. | Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, P. O. Box 1160,41080-Sevilla, Spain |
| 2. | CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France |
We prove an estimation of the Kolmogorov $\varepsilon$-entropy in $H$ of the unitary ball in the space $V$, where $H$ is a Hilbert space and $V$ is a Sobolev-like subspace of $H$. Then, by means of Zelik's result [
References:
| [1] |
A. V. Babin and M. I. Vishik,
Regular attractors of semigroups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491.
|
| [2] |
Z. Chen,
A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors, Chaos Solitons Fractals, 3 (1993), 575-582.
doi: 10.1016/0960-0779(93)90007-N. |
| [3] |
V. V. Chepyzhov and A. A. Ilyin,
A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
| [4] |
I. Dumer, M. S. Pinsker and V. V. Prelov,
On coverings of ellipsoids in Euclidean spaces, Transactions on Information Theory, 50 (2004), 2348-2356.
doi: 10.1109/TIT.2004.834759. |
| [5] |
P. Li and S. T. Yau,
On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.
|
| [6] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
| [7] |
S. Zelik,
The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mem. Math. Appl., 24 (2000), 1-25.
|
show all references
References:
| [1] |
A. V. Babin and M. I. Vishik,
Regular attractors of semigroups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491.
|
| [2] |
Z. Chen,
A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors, Chaos Solitons Fractals, 3 (1993), 575-582.
doi: 10.1016/0960-0779(93)90007-N. |
| [3] |
V. V. Chepyzhov and A. A. Ilyin,
A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819.
doi: 10.1016/S0362-546X(99)00309-0. |
| [4] |
I. Dumer, M. S. Pinsker and V. V. Prelov,
On coverings of ellipsoids in Euclidean spaces, Transactions on Information Theory, 50 (2004), 2348-2356.
doi: 10.1109/TIT.2004.834759. |
| [5] |
P. Li and S. T. Yau,
On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.
|
| [6] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2$^{nd}$ edition, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
| [7] |
S. Zelik,
The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mem. Math. Appl., 24 (2000), 1-25.
|
| [1] |
Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281 |
| [2] |
Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018 |
| [3] |
Joseph Squillace. Estimating the fractal dimension of sets determined by nonergodic parameters. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5843-5859. doi: 10.3934/dcds.2017254 |
| [4] |
V. V. Chepyzhov, A. A. Ilyin. On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 117-135. doi: 10.3934/dcds.2004.10.117 |
| [5] |
Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367 |
| [6] |
Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288 |
| [7] |
Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887 |
| [8] |
Hillel Furstenberg. From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath. Journal of Modern Dynamics, 2019, 15: 437-449. doi: 10.3934/jmd.2019027 |
| [9] |
Igor Kukavica. On Fourier parametrization of global attractors for equations in one space dimension. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 553-560. doi: 10.3934/dcds.2005.13.553 |
| [10] |
Francisco Balibrea, José Valero. On dimension of attractors of differential inclusions and reaction-diffussion equations. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 515-528. doi: 10.3934/dcds.1999.5.515 |
| [11] |
Dung Le. Exponential attractors for a chemotaxis growth system on domains of arbitrary dimension. Conference Publications, 2003, 2003 (Special) : 536-543. doi: 10.3934/proc.2003.2003.536 |
| [12] |
Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341 |
| [13] |
Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215 |
| [14] |
Dalibor Pražák, Jakub Slavík. Attractors and entropy bounds for a nonlinear RDEs with distributed delay in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1259-1277. doi: 10.3934/dcdsb.2016.21.1259 |
| [15] |
Manuel Fernández-Martínez, Miguel Ángel López Guerrero. Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1129-1137. doi: 10.3934/dcdss.2015.8.1129 |
| [16] |
Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure and Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283 |
| [17] |
Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303 |
| [18] |
G. A. Leonov. Generalized Lorenz Equations for Acoustic-Gravity Waves in the Atmosphere. Attractors Dimension, Convergence and Homoclinic Trajectories. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2253-2267. doi: 10.3934/cpaa.2017111 |
| [19] |
Manuel Fernández-Martínez. Theoretical properties of fractal dimensions for fractal structures. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1113-1128. doi: 10.3934/dcdss.2015.8.1113 |
| [20] |
Uta Renata Freiberg. Einstein relation on fractal objects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 509-525. doi: 10.3934/dcdsb.2012.17.509 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]