Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Prevalence


Authors: William Ott and James A. Yorke
Journal: Bull. Amer. Math. Soc. 42 (2005), 263-290
MSC (2000): Primary :, 28C10, 28C15, 28C20; Secondary :, 37C20, 37C45
DOI: https://doi.org/10.1090/S0273-0979-05-01060-8
Published electronically: March 30, 2005
MathSciNet review: 2149086
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Many problems in mathematics and science require the use of infinite-dimensional spaces. Consequently, there is need for an analogue of the finite-dimensional notions of `Lebesgue almost every' and `Lebesgue measure zero' in the infinite-dimensional setting. The theory of prevalence addresses this need and provides a powerful framework for describing generic behavior in a probabilistic way. We survey the theory and applications of prevalence.


References [Enhancements On Off] (What's this?)

  • 1. Proceedings of the International Congress of Mathematicians, Amsterdam, 1954. Vol. 2, Erven P. Noordhoff N. V., Groningen, 1954. MR 0070535 (16:1190j)
  • 2. Robert M. Anderson and William R. Zame, Genericity with infinitely many parameters, Adv. Theor. Econ. 1 (2001), Art. 1, 64 pp. (electronic). MR 2002579 (2004g:91038)
  • 3. A. Araujo and P. K. Monteiro, Generic nonexistence of equilibria in finance models, J. Math. Econom. 20 (1991), no. 5, 489-499. MR 1112342 (92g:90039)
  • 4. M. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 82-99. MR 0176482 (31:754)
  • 5. A. Ben-Artzi, A. Eden, C. Foias, and B. Nicolaenko, Hölder continuity for the inverse of Mañé's projection, J. Math. Anal. Appl. 178 (1993), no. 1, 22-29. MR 1231724 (94d:58091)
  • 6. B. Birnir and H. A. Hauksson, The basic attractor of the viscous Moore-Greitzer equation, J. Nonlinear Sci. 11 (2001), no. 3, 169-192. MR 1852939 (2002e:37132)
  • 7. Björn Birnir, Global attractors and basic turbulence, Nonlinear Coherent Structures in Physics and Biology (K.M. Spatschek and F.G. Mertens, eds.), NATO ASI, vol. 329, Springer-Verlag, New York, 1994.
  • 8. -, Basic attractors and basic control of nonlinear partial differential equations, Ecmi lecture notes, Chalmers University of Technology and Göteborg University, 2001.
  • 9. Björn Birnir and Rainer Grauer, An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys. 162 (1994), no. 3, 539-590. MR 1277476 (95d:58117)
  • 10. Jonathan M. Borwein and Warren B. Moors, Null sets and essentially smooth Lipschitz functions, SIAM J. Optim. 8 (1998), no. 2, 309-323 (electronic). MR 1618798 (99g:49013)
  • 11. Jens Peter Reus Christensen, On sets of Haar measure zero in abelian Polish groups, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel J. Math. 13 (1972), 255-260 (1973). MR 0326293 (48:4637)
  • 12. Darrell Duffie and William Zame, The consumption-based capital asset pricing model, Econometrica 57 (1989), no. 6, 1279-1297. MR 1035113 (91c:90015)
  • 13. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988, General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication. MR 1009162 (90g:47001a)
  • 14. Kenneth Falconer, Fractal geometry, John Wiley & Sons Ltd., Chichester, 1990, Mathematical foundations and applications. MR 1102677 (92j:28008)
  • 15. Michael Field, Ian Melbourne, and Matthew Nicol, Symmetric attractors for diffeomorphisms and flows, Proc. London Math. Soc. (3) 72 (1996), no. 3, 657-696. MR 1376773 (97d:58125)
  • 16. C. Foias and E. Olson, Finite fractal dimension and Hölder-Lipschitz parametrization, Indiana Univ. Math. J. 45 (1996), no. 3, 603-616. MR 1422098 (97m:58120)
  • 17. Peter K. Friz and James C. Robinson, Smooth attractors have zero ``thickness'', J. Math. Anal. Appl. 240 (1999), no. 1, 37-46. MR 1728206 (2001f:37130)
  • 18. -, Parametrising the attractor of the two-dimensional Navier-Stokes equations with a finite number of nodal values, Phys. D 148 (2001), no. 3-4, 201-220. MR 1820361 (2002a:35023)
  • 19. Jean-Michel Ghidaglia, Martine Marion, and Roger Temam, Generalization of the Sobolev-Lieb-Thirring inequalities and applications to the dimension of attractors, Differential Integral Equations 1 (1988), no. 1, 1-21. MR 0920485 (89d:35137)
  • 20. Brian R. Hunt, The prevalence of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 122 (1994), no. 3, 711-717. MR 1260170 (95d:26009)
  • 21. Brian R. Hunt and Vadim Yu. Kaloshin, How projections affect the dimension spectrum of fractal measures, Nonlinearity 10 (1997), no. 5, 1031-1046. MR 1473372 (98k:28010)
  • 22. -, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999), no. 5, 1263-1275. MR 1710097 (2001a:28009)
  • 23. Brian R. Hunt, Judy A. Kennedy, Tien-Yien Li, and Helena E. Nusse, SLYRB measures: natural invariant measures for chaotic systems, Phys. D 170 (2002), no. 1, 50-71. MR 1945459 (2004d:37033)
  • 24. Brian R. Hunt, Tim Sauer, and James A. Yorke, Prevalence: a translation-invariant ``almost every'' on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217-238. MR 1161274 (93k:28018)
  • 25. -, Prevalence. An addendum to: ``Prevalence: a translation-invariant `almost every' on infinite-dimensional spaces'' [Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217-238; MR 1161274 (93k:28018)], Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 306-307. MR 1191479 (93k:28019)
  • 26. V. Yu. Kaloshin, Prevalence in spaces of finitely smooth mappings, Funktsional. Anal. i Prilozhen. 31 (1997), no. 2, 27-33, 95. MR 1475321 (98i:58040)
  • 27. -, Some prevalent properties of smooth dynamical systems, Tr. Mat. Inst. Steklova 213 (1997), Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 123-151. MR 1632241 (99h:58100)
  • 28. Vadim Yu. Kaloshin, An extension of the Artin-Mazur theorem, Ann. of Math. (2) 150 (1999), no. 2, 729-741. MR 1726706 (2000j:37020)
  • 29. -, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys. textbf211 (2000), no. 1, 253-271. MR 1757015 (2001e:37035)
  • 30. Vadim Yu. Kaloshin and Brian R. Hunt, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms. I, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 17-27 (electronic). MR 1826992 (2002d:37032)
  • 31. Robert Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153-155. MR 0248779 (40:2030)
  • 32. E. Lieb and W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics, Princeton University Press, 1976.
  • 33. Ricardo Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 230-242. MR 0654892 (84k:58119)
  • 34. J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3) 4 (1954), 257-302. MR 0063439 (16:121g)
  • 35. Pertti Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 227-244. MR 0409774 (53:13526)
  • 36. Ian Melbourne and Ian Stewart, Symmetric $\omega$-limit sets for smooth $\Gamma$-equivariant dynamical systems with $\Gamma\sp 0$ abelian, Nonlinearity 10 (1997), no. 6, 1551-1567. MR 1483554 (98k:58169)
  • 37. John Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), no. 2, 177-195. MR 0790735 (87i:58109a)
  • 38. T. Okon, Dimension estimate preserving embeddings for compacta in metric spaces, Arch. Math. (Basel) 78 (2002), no. 1, 36-42. MR 1887314 (2002m:60009)
  • 39. William Ott, Brian Hunt, and Vadim Kaloshin, The effect of projections on fractal sets and measures in Banach spaces, submitted to Ergodic Theory & Dynamical Systems, 2004.
  • 40. William Ott and James A. Yorke, Learning about reality from observation, SIAM J. Appl. Dyn. Syst. 2 (2003), no. 3, 297-322 (electronic). MR 2031277 (2004m:37041)
  • 41. J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560-566. MR 0000097 (1:18e)
  • 42. Yuval Peres and Wilhelm Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102 (2000), no. 2, 193-251. MR 1749437 (2001d:42013)
  • 43. James C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001, An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888 (2003f:37001a)
  • 44. Tim Sauer, James A. Yorke, and Martin Casdagli, Embedology, J. Statist. Phys. 65 (1991), no. 3-4, 579-616. MR 1137425 (93c:58147)
  • 45. Timothy D. Sauer and James A. Yorke, Are the dimensions of a set and its image equal under typical smooth functions?, Ergodic Theory Dynam. Systems 17 (1997), no. 4, 941-956. MR 1468109 (98f:58127)
  • 46. E. V. Shchepin and D. Repovs, On smoothness of compacta, J. Math. Sci. (New York) 100 (2000), no. 6, 2716-2726, Pontryagin Conference, 2, Nonsmooth Analysis and Optimization (Moscow, 1998). MR 1778991 (2001f:58005)
  • 47. Barry Simon, Operators with singular continuous spectrum. I. General operators, Ann. of Math. (2) 141 (1995), no. 1, 131-145. MR 1314033 (96a:47038)
  • 48. Masato Tsujii, Fat solenoidal attractors, Nonlinearity 14 (2001), no. 5, 1011-1027. MR 1862809 (2002j:37035)
  • 49. Hassler Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645-680. MR 1503303
  • 50. Lai-Sang Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys. 108 (2002), no. 5-6, 733-754, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933431 (2003g:37042)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): :, 28C10, 28C15, 28C20, :, 37C20, 37C45

Retrieve articles in all journals with MSC (2000): :, 28C10, 28C15, 28C20, :, 37C20, 37C45


Additional Information

William Ott
Affiliation: Courant Institute of Mathematical Sciences, New York, New York 10012
Email: ott@cims.nyu.edu

James A. Yorke
Affiliation: Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Email: yorke@ipst.umd.edu

DOI: https://doi.org/10.1090/S0273-0979-05-01060-8
Keywords: Prevalence
Received by editor(s): August 11, 2004
Published electronically: March 30, 2005
Additional Notes: This work is based on an invited talk given by the authors in January 2004 at the annual meeting of the AMS in Phoenix, AZ
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society