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Lipschitz deviation and embeddings of global attractors. (English) Zbl 1213.37112

The authors study the regularity of embeddings of finite-dimensional compact subsets of an infinite-dimensional Hilbert space into an Euclidean space.
B. R. Hunt and V. Y. Kaloshin [Nonlinearity 12, No.5, 1263–1275 (1999; Zbl 0932.28006)] have proved the following
Theorem. Let \(X\) be a compact subset of a real Hilbert space \(H\) with upper box-counting dimension \(\dim_{box}(X) = d\) and let \(\tau(X)\) be the thickness exponent of \(X\). Let \(N > 2d\) be an integer and let \(\alpha\) be a real number with \[ 0<\alpha < \frac{N-2d}{N(1+\tau(X)/2)}. \eqno{(1)} \] Then for a prevalent set of linear maps \(L : H \to \mathbb{R}^N\) there exists a \(C > 0\) such that \[ C|L(x) - L(y)|^{\alpha}\geq \|x - y\| \quad \text{for all} \quad x, y \in X; \eqno{(2)} \] in particular these maps are injective on \(X\).
Here the thickness exponent is used to bound explicitly the Hőlder exponent of the inverse of \(L\) restricted to the image of \(X\). The regularity of the inverse of such a linear map is important for understanding how it may distort a compact set, despite being injective on it.
More later W. Ott, B. Hunt and V. Kaloshin [Ergodic Theory Dyn. Syst. 26, No. 3, 869–891 (2006; Zbl 1123.28012)] also utilized the thickness exponent to show how the Hausdorff dimension of a subset of an infinite-dimensional space is affected by mappings into finite-dimensional spaces. The Hausdorff dimension of the set \(X\) is defined by \[ \dim_H(X) =\inf {\{d :\mathcal{H}^d(X) = 0\}} = \sup{\{d : \mathcal{H}^d(X) = \infty\}}, \] where \(\mathcal{H}^d\) is the \(d\)-dimensional Hausdorff measure \[ \mathcal{H}^d(X) = \lim _{\delta \to 0}\left ( \inf \left\{ \sum\limits_{i=1}^{\infty}|U_i|^d : \{U_i\}\;\text{ is a}\quad\delta-\text{cover of}\quad X \right\}\right) \]
Here \(|U_i|=\sup {\{|x-y|: x,y \in U_i\}}\), is the diameter of the set \(U\), and a \(\delta\)-cover is a cover \(\{U_i\}\), such that \(|U_i|\leq \delta\) for all \(i\).
It follows from standard properties of the Hausdorff dimension that for any bounded linear map \(L\) that satisfies (3) \[ \frac{N-2d}{N(1+\tau(X)/2)}\dim_H(X)\leq \dim_H(L(X)) \leq \dim_H(X) \eqno{(3)} \] The main result of this article consists in the Theorem:
Let \(H\) be a real Hilbert space. Let \(X \subset H\) be a compact set with thickness \(\tau(X)\). For a prevalent set of linear maps \(L : H \to \mathbb{R}^N,\) \[ \dim_H(L(X))\geq \min \left\{N, \frac{\dim_H(X)}{1+\tau(X)/2}\right\}. \]
In the reviewed article authors introduce a variant of the thickness exponent, the Lipschitz deviation \(\text{dev}(X)\), show that in both of the above results this notion can be used in place of the thickness exponent, and (appealing to results from the theory of approximate inertial manifolds) prove that \(\text{dev} (X)= 0\) for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt and Kaloshin. In particular, \(\text{dev}(X) = 0\) for the attractor of the \(2D\) Navier-Stokes equations with forcing \(f\in L^2\), while current results only guarantee that \(\tau(X) = 0\), when \(f\in C^{\infty}\).

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B42 Inertial manifolds
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
35B41 Attractors
35G25 Initial value problems for nonlinear higher-order PDEs
35Q30 Navier-Stokes equations
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
54F45 Dimension theory in general topology
57N35 Embeddings and immersions in topological manifolds
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