| Full text | |
| Author(s): |
Carvalho, Alexandre N.
;
Cunha, Arthur C.
;
Langa, Jose A.
;
Robinson, James C.
Total Authors: 4
|
| Document type: | Journal article |
| Source: | Journal of Mathematical Analysis and Applications; v. 509, n. 2, p. 21-pg., 2022-05-15. |
| Abstract | |
We give a simple proof of a result due to Mane (1981) [17] that a compact subset A of a Banach space that is negatively invariant for a map S is finite-dimensional if DS(x) = C(x) + L(x), where C is compact and L is a contraction (and both are linear). In particular, we show that if S is compact and differentiable then A is finite-dimensional. We also prove some results (following Malek et al. (1994) [15] and Zelik (2000) [23]) that give bounds on the (box-counting) dimension of such sets assuming a 'smoothing property': in its simplest form this requires S to be Lipschitz from X into another Banach space Z that is compactly embedded in X. The resulting bounds depend on the Kolmogorov epsilon-entropy of the embedding of Z into X. We give applications to an abstract semilinear parabolic equation and the two-dimensional Navier-Stokes equations on a periodic domain.(c) 2022 Elsevier Inc. All rights reserved. (AU) | |
| FAPESP's process: | 16/26289-5 - Estimates of the Fractal Dimension of Attractors for Autonomous and Non-Autonomous Dynamical Systems |
| Grantee: | Arthur Cavalcante Cunha |
| Support Opportunities: | Scholarships in Brazil - Doctorate |
| FAPESP's process: | 20/14075-6 - Dynamical systems and their attractors under perturbations |
| Grantee: | Alexandre Nolasco de Carvalho |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 18/10634-0 - Estimates of the fractal dimension of attractors for autonomous and non-autonomous dynamical systems: applications |
| Grantee: | Arthur Cavalcante Cunha |
| Support Opportunities: | Scholarships abroad - Research Internship - Doctorate |