# Smooth Attractors Have Zero “Thickness”☆

@article{Friz1999SmoothAH, title={Smooth Attractors Have Zero “Thickness”☆}, author={Peter K. Friz and James C. Robinson}, journal={Journal of Mathematical Analysis and Applications}, year={1999}, volume={240}, pages={37-46} }

Abstract A finite-dimensional global attractor A can be embedded, using some linear map L , into a Euclidean space R k of sufficiently high dimension. The Holder exponent of L − 1 depends upon k and upon τ( A ), the “thickness exponent” of A . We show that global attractors which are uniformly bounded in the Sobolev spaces H s for all s > 0 have τ( A ) = 0. It follows, using a result of B. R. Hunt and V. Y. Kaloshin, that the Holder constant of the inverse of a typical linear embedding into R…

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## References

SHOWING 1-10 OF 31 REFERENCES

Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces

- Mathematics
- 1992

We present a measure-theoretic condition for a property to hold «almost everywhere» on an infinite-dimensional vector space, with particularemphasis on function spaces such as C k and L p . Like the…

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

- PhysicsSIAM Rev.
- 1991

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, and An interpretation of classical Yang-Mills theory, Cambridge Univ.

Exponential Attractors for Dissipative Evolution Equations

- Mathematics
- 1995

Construction of Exponential Attractors for Maps Exponential Attractors for Dissipative Evolution Equations of First Order Approximation of Exponential Attractors Applications Exponential Attractors…

Fractal geometry

- Psychology
- 1989

Editor's note: The following articles by Steven G. Krantz and Benoit B. Mandelbrot have an unusual history. In the fall of 1988, Krantz asked the Bulletin of the American Mathematical Society Book…

On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions

- Mathematics, Computer Science
- 1993

Different definitions of the correlation dimension are considered and some relationships between them and other characteristics of dimension type such as Hausdorff dimension, box dimension, etc are found.

PARTIAL DIFFERENTIAL EQUATIONS

- Mathematics
- 1941

Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear…

Prevalence: an addendum

- Mathematics
- 1993

The authors mention work of Christensen, Mycielski, Tsujii, and others which is closely related to a survey article by the first author [math.FA/9210220].