Finite-dimensional negatively invariant subsets of Banach spaces

Author(s)

Carvalho, Alexandre N. ; Cunha, Arthur C. ; Langa, Jose A. ; Robinson, James C.

• 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