Lipschitz deviation and embeddings of global attractors

Abstract

Kaloshin (1999 Nonlinearity 12 1263-75) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d < k into R N for any N > 2k + 1, using a linear map L whose inverse is Hölder continuous with exponent α < (N − 2d)/N (1 + τ (X)/2), where τ (X) is the 'thickness exponent' of X. More recently, Ott et al (2006 Ergod. Theory Dyn. Syst. 26 869-91) studied the effect of such embeddings on the Hausdorff dimension of X, and showed that for 'most' linear mapsThey also conjectured that 'many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero'. In this paper we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X): we show that in both of the above results this can be used in place of the thickness exponent, and-appealing to results from the theory of approximate inertial manifolds-we prove that 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, dev(X) = 0 for the attractor of the 2D Navier-Stokes equations with forcing f ∈ L 2 , while current results only guarantee that τ (X) = 0, when f ∈ C ∞ .

Attractors and Finite-Dimensional Behaviour in the 2D Navier-Stokes Equations

“…One can deduce from this [81,107] that the Lipschitz deviation of the Navier-Stokes attractor is zero, even when we only have ∈ . Proposition 43.…”

“…The Lipschitz deviation, used in the statement of the theorem given here, was introduced by Olson & Robinson [80], with the definition revised and investigated further by Pinto de Moura & Robinson [81].…”

Контрпримеры К Регулярности Проекций Мане В Теории Аттракторов

“…Поэтому редуцированная ДС (1.2) также оказывается непрерывной по Гёльдеру, и, при некоторых естественных дополнительных условиях на исходную диссипативную систему, константа Гёльдера α может быть сделана сколь угодно близкой к единице за счет увеличения размерности N (см. [12], [26], [29] и цитируемую там литературу). Также известно, что редуцированная ДС S(t) может быть описана системой ОДУ (1.3) в R N для некоторого непрерывного по Гёльдеру на P A векторного поля F (см.…”

Observing Lyapunov Exponents of Infinite-Dimensional Dynamical Systems

“…Theorems 1.3 and 1.5 remain true when one replaces the thickness exponent of A with the Lipschitz deviation dev( A ) [22]. Roughly speaking, τ ( A ) measures how well A can be approximated by finite-dimensional subspaces of H , while dev( A ) measures how well A can be approximated by the graphs of Lipschitz functions defined on finite-dimensional subspaces of H (with lower values of τ ( A ) and dev( A ) indicating better approximability).…”