Finite-dimensional negatively invariant subsets of Banach spaces
The box-counting dimension (also called fractal dimension or limit capacity) of a compact set \(A\), when finite, allows one to prove the finiteness of both the Hausdorff dimension and the topological dimension. Furthermore it can be used to prove that one can injectively project \(A\) into an Euclidean space. In [\textit{R. Mañé}, Lect. Notes Math. 898, 230--242 (1981; Zbl 0544.58014)] we learned that if \(A\) is a compact subset of a Banach space \(X\) that is negatively invariant for a map \(S\), meaning that \(S(A)\supset A\), and for each \(x\in A\) we can decompose the derivative \(DS(x)=C(x)+L(x)\), where \(C(x)\) is a compact linear operator and \(L\) is a linear strict contraction, then the box-counting dimension of \(A\) is finite. The authors begin presenting a rather simple proof of this result and, in particular, they obtain that when \(S\) is a differentiable compact map, then \(A\) has finite box-counting dimension. Furthermore, they obtain bounds for the box-counting dimension of such sets following the works [\textit{J. Málek} et al., Acta Appl. Math. 37, No. 1--2, 83--97 (1994; Zbl 0815.35084); \textit{S. V. Zelik}, Math. Notes 67, No. 2, 248--251 (2000; Zbl 0974.35016); translation from Mat. Zametki 67, No. 2, 304--308 (2000)]. To this end they assume some smoothing properties of \(S\), which in their simplest form require \(S\) to be Lipschitz from \(X\) into another Banach space \(Z\) that is compactly embedded in \(X\). The resulting bounds depend on the Kolmogorov \(\varepsilon\)-entropy of the embedding of \(Z\) into \(X\). They apply their results to an abstract semilinear parabolic equation and a two-dimensional Navier-Stokes equation on a periodic domain.
scientific article; zbMATH DE number 7472982
scientific article