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Embedding of global attractors and their dynamics


Authors: Eleonora Pinto de Moura, James C. Robinson and J. J. Sánchez-Gabites
Journal: Proc. Amer. Math. Soc. 139 (2011), 3497-3512
MSC (2010): Primary 37L30, 54H20, 57N60
DOI: https://doi.org/10.1090/S0002-9939-2011-10759-7
Published electronically: February 9, 2011
MathSciNet review: 2813382
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Abstract: Suppose that $ \mathcal A$ is the global attractor associated with a dissipative dynamical system on a Hilbert space $ H$.

If the set $ \mathcal A-\mathcal A$ has finite Assouad dimension $ d$, then for any $ m>d$ there are linear homeomorphisms $ L:\mathcal{A}\rightarrow\mathbb{R}^{m+1}$ such that $ L{\mathcal A}$ is a cellular subset of $ \mathbb{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 $ \mathbb{R}^k$ is the global attractor of some smooth ordinary differential equation on $ \mathbb{R}^k$ if and only if it is cellular, and hence we obtain a dynamical system on $ \mathbb{R}^k$ for which $ L{\mathcal A}$ is the global attractor. However, $ L\mathcal{A}$ consists entirely of stationary points.

In order for the dynamics on $ L\mathcal A$ to reproduce those on $ L\mathcal A$ we need to make an additional assumption, namely that the dynamics restricted to $ \mathcal 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 $ \mathbb{R}^N$). Given this we can construct an ordinary differential equation in some $ \mathbb{R}^k$ (where $ k$ is determined by $ d$ and $ \alpha$) that has unique solutions and reproduces the dynamics on $ \mathcal A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $ \mathcal X$ arbitrarily close to $ L\mathcal A$.


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  • 1. P. Assouad. Plongements Lipschitziens dans $ \mathbb{R}^N$. Bull. Soc. Math. France, 111:429-448, 1983. MR 763553 (86f:54050)
  • 2. J.E. Billotti and J.P. LaSalle. Dissipative periodic processes. Bull. Amer. Math. Soc. 77:1082-1088, 1971. MR 0284682 (44:1906)
  • 3. K. Borsuk. Theory of Shape, volume 59 of Monografie Matematyczne. Polish Scientific Publishers, Warszawa, 1975. MR 0418088 (54:6132)
  • 4. M. Brown. A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc., 66:74-76, 1960. MR 0117695 (22:8470b)
  • 5. V.V. Chepyzhov and M.I. Vishik. Attractors for equations of mathematical physics, volume 49 of American Mathematical Society Colloquium Publications. American Mathematical Society, 2002. MR 1868930 (2003f:37001c)
  • 6. J.P.R. Christensen. Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings. Publ. Dép. Math. (Lyon), 10(2):29-39, 1973. MR 0361770 (50:14215)
  • 7. P. Constantin and C. Foias. Navier-Stokes Equations. University of Chicago Press, Chicago, 1988. MR 972259 (90b:35190)
  • 8. P. Constantin, C. Foias, B. Nicolaenko and R. Temam. Integral manifolds and inertial manifolds for dissipative partial differential equations. Springer-Verlag, New York, 1989. MR 966192 (90a:35026)
  • 9. R.J. Daverman. Decomposition of manifolds. Academic Press Inc., London, 1986. MR 872468 (88a:57001)
  • 10. C.H. Dowker. On countably paracompact spaces. Canad. J. Math., 3:219-224, 1951. MR 0043446 (13:264c)
  • 11. A. Eden, C. Foias, B. Nicolaenko and R. Temam. Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics Series. John Wiley and Sons, New York, 1994. MR 1335230 (96i:34148)
  • 12. C. Foias and E.J. Olson. Finite fractal dimension and Hölder-Lipschitz parametrization. Indiana Univ. Math. J., 45:603-616, 1996. MR 1422098 (97m:58120)
  • 13. C. Foias, O. Manley and R. Temam. Modelling of the interaction of small and large eddies in two-dimensional turbulent flows. RAIRO Modél. Math. Anal. Numér., 22(1):93-118, 1988. MR 934703 (89h:76022)
  • 14. C. Foias, G.R. Sell and R. Temam. Variétés inertielles des équations différentielles dissipatives. C. R. Acad. Sci. Paris I, 301:139-141, 1985. MR 801946 (87f:35214a)
  • 15. C. Foias, G.R. Sell and R. Temam. Inertial manifolds for nonlinear evolutionary equations. J. Differential Equations, 73:309-353, 1988. MR 943945 (89e:58020)
  • 16. B. M. Garay. Strong cellularity and global asymptotic stability. Fund. Math., 138:147-154, 1991. MR 1124542 (92h:54053)
  • 17. R. Geoghegan and R.R. Summerhill. Concerning the shapes of finite-dimensional compacta. Trans. Amer. Math. Soc., 179:281-292, 1973. MR 0324637 (48:2987)
  • 18. B. Günther. Construction of differentiable flows with prescribed attractor. Topology Appl., 62:87-91, 1995. MR 1318430 (96e:58095)
  • 19. B. Günther and J. Segal. Every attractor of a flow on a manifold has the shape of a finite polyhedron. Proc. Amer. Math. Soc, 119:321-329, 1993. MR 1170545 (93k:54044)
  • 20. P. Hartman. Ordinary Differential Equations. John Wiley and Sons, 1964. MR 0171038 (30:1270)
  • 21. B.R. Hunt and V.Y. Kaloshin. Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity, 12:1263-1275, 1999. MR 1710097 (2001a:28009)
  • 22. B.R. Hunt, T. Sauer and J.A. Yorke. Prevalence: A translation-invariant `almost every' on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.), 27(2):217-238, 1992. MR 1161274 (93k:28018)
  • 23. W. Hurewicz and H. Wallman. Dimension Theory. Princeton University Press, Princeton, NJ, 1948. MR 0006493 (3:312b)
  • 24. L. Kapitanski and I. Rodnianski. Shape and Morse theory of attractors. Comm. Pure Appl. Math., 53(2):218-242, 2000. MR 1721374 (2000h:37019)
  • 25. R.C. Kirby and L.C. Siebenmann. Foundational Essays on Topological Manifolds, Smoothings and Triangulations. Annals of Mathematical Studies, 88. Princeton University Press, Princeton, NJ, 1977. MR 0645390 (58:31082)
  • 26. I. Kukavica. Log-log convexity and backward uniqueness. Proc. Amer. Math. Soc. 135:2415-2421, 2007. MR 2302562 (2009a:35119)
  • 27. O. Ladyzhenskaya. Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge, 1991. MR 1133627 (92k:58040)
  • 28. J. Luukkainen. Assouad dimension: Antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc., 35(1):23-76, 1998. MR 1608518 (99m:54023)
  • 29. R. Mañé. On the dimension of the compact invariant sets of certain nonlinear maps. Springer Lecture Notes in Math., 898:230-242, 1981. MR 654892 (84k:58119)
  • 30. S. Mardesić and J. Segal. Shape Theory. North-Holland, 1982. MR 676973 (84b:55020)
  • 31. E.J. McShane. Extension of the range of functions. Bull. Amer. Math. Soc. 40:837-842, 1934. MR 1562984
  • 32. A. Norton and C. Pugh. Critical sets in the plane. Michigan Math. J., 38:441-459, 1991. MR 1116500 (92f:57032)
  • 33. E.J. Olson. Bouligand dimension and almost Lipschitz embeddings. Pacific J. Math., 202:459-474, 2002. MR 1888175 (2003a:37030)
  • 34. E.J. Olson and J.C. Robinson. Almost bi-Lipschitz embeddings and almost homogeneous sets. Trans. Amer. Math. Soc., 362(1):145-168, 2010. MR 2550147
  • 35. E. Pinto de Moura and J.C. Robinson. Orthogonal sequences and regularity of embeddings into finite-dimensional spaces. J. Math. Anal. Appl. 368:254-262, 2010a. MR 2609274
  • 36. E. Pinto de Moura and J.C. Robinson. Log-Lipschitz continuity of the vector field on the attractor of certain parabolic equations. Submitted, 2010b.
  • 37. J.C. Robinson. Global Attractors: Topology and finite-dimensional dynamics. J. Dynam. Differential Equations, 11(3):557-581, 1999. MR 1693866 (2000k:37123)
  • 38. J.C. Robinson. Infinite-dimensional dynamical systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001. MR 1881888 (2003f:37001a)
  • 39. J.C. Robinson. Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces. Nonlinearity, 22:711-728, 2009. MR 2486352
  • 40. J.C. Robinson. Dimensions, embeddings, and attractors, Cambridge Tracts in Math., 186. Cambridge University Press, Cambridge, 2011.
  • 41. A.V. Romanov. Finite-dimensional limiting dynamics for dissipative parabolic equations. Sb. Math., 191(3):415-429, 2000. MR 1773256 (2001f:37133)
  • 42. J.M.R. Sanjurjo. On the structure of uniform attractors. J. Math. Anal. Appl., 192:519-528, 1995. MR 1332224 (96c:58109)
  • 43. E.M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ, 1982. MR 0290095 (44:7280)
  • 44. R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Springer Applied Mathematical Sciences. Springer-Verlag, Berlin, 2nd edition, 1997. MR 1441312 (98b:58056)

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Additional Information

Eleonora Pinto de Moura
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: epmoura@ime.unicamp.br

James C. Robinson
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: j.c.robinson@warwick.ac.uk

J. J. Sánchez-Gabites
Affiliation: Mathematical Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: j.j.sanchez-gabites@warwick.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2011-10759-7
Received by editor(s): August 12, 2010
Received by editor(s) in revised form: August 23, 2010
Published electronically: February 9, 2011
Additional Notes: The first author is sponsored by CAPES and would like to thank CAPES for all their support during her Ph.D
The second and third authors are supported by EPSRC Leadership Fellowship EP/G007470/1.
Communicated by: Professor Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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