Bouligand dimension and almost Lipschitz embeddings

Trans. Amer. Math. Soc.

2009

Abstract. This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (biLipschitz to within logarithmic corrections). The image of this set is no longer homogeneous, but 'almost homogeneous'. We therefore study the problem of embedding an almost homogeneous subset X of a Hilbert space H into a finite-dimensional Euclidean space. We show that if X is a compact subset of a Hilbert space and X − X is almost homogeneous, then, for N sufficiently large, a prevalent set of linear maps from X into R N are almost bi-Lipschitz between X and its image.

Section: Aside: Compact Spaces and Local Versions Of (Almost) Homogensupporting

confidence: 68%

Section: Almost Bi-lipschitz Embeddingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Almost bi-Lipschitz embeddings and almost homogeneous sets

Olson

Trans. Amer. Math. Soc.

2009

“…Even then, dim A (A) < ∞ still does not imply that dim A (A − A) < ∞ (see Olson (2002)). The only attractors known to satisfy this condition are those that are subsets of inertial manifolds; but, of course, in this case, a finitedimensional system of ODEs that reproduces the behaviour on A is already known to exist, making this class of examples of little interest.…”

Section: Embedding the Dynamics On A Into Euclidean Spacementioning

confidence: 99%

Embedding of global attractors and their dynamics

Moura

Sánchez-Gabites

2011

Proc. Amer. Math. Soc.

Abstract. Suppose that A is the global attractor associated with a dissipative dynamical system on a Hilbert space H.If the set A − A has finite Assouad dimension d, then for any m > d there are linear homeomorphisms L : A → R m+1 such that LA is a cellular subset of R m+1 and L −1 is log-Lipschitz (i.e. Lipschitz to within logarithmic corrections). We give a relatively simple proof that a compact subset X of R k is the global attractor of some smooth ordinary differential equation on R k if and only if it is cellular, and hence we obtain a dynamical system on R k for which LA is the global attractor. However, LA consists entirely of stationary points.In order for the dynamics on LA to reproduce those on LA we need to make an additional assumption, namely that the dynamics restricted to A are generated by a log-Lipschitz continuous vector field (this appears overly restrictive when H is infinite-dimensional, but is clearly satisfied when the initial dynamical system is generated by a Lipschitz ordinary differential equation on R N ). Given this we can construct an ordinary differential equation in some R k (where k is determined by d and α) that has unique solutions and reproduces the dynamics on A. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor X arbitrarily close to LA.

“…where N (r, ρ) is the number of ρ-balls required to cover any r-ball in A (see Olson [11] for proof). For a more comprehensive treatment of the Assouad dimension see Luukkainen [8].…”

Section: Theorem 16 (See Robinson [14]) Let X Be a Compact Subset mentioning

confidence: 99%

Orthogonal sequences and regularity of embeddings into finite-dimensional spaces

Moura

Journal of Mathematical Analysis and Applications

2010

This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c 0 of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finitedimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X − X has finite Assouad dimension.