Abstract
In this chapter we investigate the dimension of attractors for autonomous and non-autonomous problems. The treatment is necessarily abstract since application of the results generally makes use of certain differentiability properties that need to be checked carefully in each particular application. We will apply the results of this chapter to an abstract semilinear parabolic equation in Chap. 6, to the two-dimensional Navier–Stokes equations in Chap. 11, and to a non-autonomous version of the Chafee–Infante equation in Chap. 13.
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Notes
- 1.
The lim sup is necessary since there are examples for which the limit itself does not exist (see Robinson 2011). One can also define the lower box-counting dimension replacing the lim sup by a lim inf, but the upper box-counting dimension is to be preferred since this stronger definition is required in the proof of Theorem 4.22.
- 2.
If A(t) is a compact invariant set for a process S( ⋅, ⋅), then we can consider the discrete family of sets K j = A(j) and corresponding maps f j = S(j + 1, j), \(j \in \mathbb{Z}\), so that K j + 1 = f j (K j ). If we can show that each K j has finite box-counting dimension, we can use the fact that dimB is non-increasing under Lipschitz maps to ‘fill in the gaps’ and bound the dimension of A(t) for \(t\notin \mathbb{Z}\).
- 3.
In fact Mañé’s proof requires only that dim(V ) > dimH(X − X), where dimH denotes the Hausdorff dimension. For an embedding result along the lines of our Theorem 4.22 under this assumption see Theorem 6.2 in Robinson (2011), where it is also shown that this condition is not strong enough to guarantee any particular form for the modulus of continuity of the inverse of L.
- 4.
In fact, Hunt and Kaloshin (1999) defined the thickness exponent of K to be
$$\tau (K) ={ limsup }_{\varepsilon \rightarrow 0}\frac{\log d(K,\varepsilon )} {-\log \varepsilon },$$where d(K, ε) is the smallest dimension of a linear subspace V of H such that dist(K, V ) ≤ ε; this generally provides a much lower bound on d n . The argument here shows that in general τ(K) ≤ dimB(K), but Friz and Robinson (1999) showed that if K ⊂ L 2 consists of smooth functions (bounded in H k for every \(k \in \mathbb{N}\)), then in fact τ(K) = 0. For more on this and other thickness exponents see Chap. 7 in Robinson (2011).
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Carvalho, A.N., Langa, J.A., Robinson, J.C. (2013). Finite-dimensional attractors. In: Attractors for infinite-dimensional non-autonomous dynamical systems. Applied Mathematical Sciences, vol 182. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4581-4_4
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