Finite-dimensional negatively invariant subsets of Banach spaces

Abstract

We give a simple proof of a result due to Mañé (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 Málek 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 ε-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.