Finite-dimensional negatively invariant subsets of Banach spaces

We give a simple proof of result due to Mañé (1981) [17] that compact subset A Banach space is negatively invariant for map S finite-dimensional if D ( x ) = C + L , where and contraction (and both are linear). In particular, we show differentiable then finite-dimensional. also prove some results (following Málek et al. (1994) [15] Zelik (2000) [23] bounds on the (box-counting) dimension such sets assuming ‘smoothing property’: in its simplest form this requires be Lipschitz from X into another Z compactly embedded . The resulting depend Kolmogorov ? -entropy embedding applications an abstract semilinear parabolic equation two-dimensional Navier–Stokes equations periodic domain.