## Prevalence

HTML articles powered by AMS MathViewer

- by William Ott and James A. Yorke PDF
- Bull. Amer. Math. Soc.
**42**(2005), 263-290 Request permission

## 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

*Proceedings of the International Congress of Mathematicians, Amsterdam, 1954. Vol. 2*, P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1954. MR**0070535**- Robert M. Anderson and William R. Zame,
*Genericity with infinitely many parameters*, Adv. Theor. Econ.**1**(2001), Art. 1, 64. MR**2002579**, DOI 10.2202/1534-5963.1003 - A. Araujo and P. K. Monteiro,
*Generic nonexistence of equilibria in finance models*, J. Math. Econom.**20**(1991), no. 5, 489–499. MR**1112342**, DOI 10.1016/0304-4068(91)90005-E - M. Artin and B. Mazur,
*On periodic points*, Ann. of Math. (2)**81**(1965), 82–99. MR**176482**, DOI 10.2307/1970384 - 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**, DOI 10.1006/jmaa.1993.1288 - 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**, DOI 10.1007/s00332-001-0310-2
Birnir1994a Björn Birnir, - 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** - Jonathan M. Borwein and Warren B. Moors,
*Null sets and essentially smooth Lipschitz functions*, SIAM J. Optim.**8**(1998), no. 2, 309–323. MR**1618798**, DOI 10.1137/S1052623496305213 - Jens Peter Reus Christensen,
*On sets of Haar measure zero in abelian Polish groups*, Israel J. Math.**13**(1972), 255–260 (1973). MR**326293**, DOI 10.1007/BF02762799 - Darrell Duffie and William Zame,
*The consumption-based capital asset pricing model*, Econometrica**57**(1989), no. 6, 1279–1297. MR**1035113**, DOI 10.2307/1913708 - 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** - Kenneth Falconer,
*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677** - 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**, DOI 10.1112/plms/s3-72.3.657 - C. Foias and E. Olson,
*Finite fractal dimension and Hölder-Lipschitz parametrization*, Indiana Univ. Math. J.**45**(1996), no. 3, 603–616. MR**1422098**, DOI 10.1512/iumj.1996.45.1326 - Peter K. Friz and James C. Robinson,
*Smooth attractors have zero “thickness”*, J. Math. Anal. Appl.**240**(1999), no. 1, 37–46. MR**1728206**, DOI 10.1006/jmaa.1999.6569 - Peter K. Friz and James C. Robinson,
*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**, DOI 10.1016/S0167-2789(00)00179-2 - 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**920485** - Brian R. Hunt,
*The prevalence of continuous nowhere differentiable functions*, Proc. Amer. Math. Soc.**122**(1994), no. 3, 711–717. MR**1260170**, DOI 10.1090/S0002-9939-1994-1260170-X - 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**, DOI 10.1088/0951-7715/10/5/002 - Brian R. Hunt and Vadim Yu. Kaloshin,
*Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces*, Nonlinearity**12**(1999), no. 5, 1263–1275. MR**1710097**, DOI 10.1088/0951-7715/12/5/303 - 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**, DOI 10.1016/S0167-2789(02)00445-1 - 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**, DOI 10.1090/S0273-0979-1992-00328-2 - 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**, DOI 10.1090/S0273-0979-1992-00328-2 - V. Yu. Kaloshin,
*Prevalence in spaces of finitely smooth mappings*, Funktsional. Anal. i Prilozhen.**31**(1997), no. 2, 27–33, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl.**31**(1997), no. 2, 95–99. MR**1475321**, DOI 10.1007/BF02466014 - V. Yu. Kaloshin,
*Some prevalent properties of smooth dynamical systems*, Tr. Mat. Inst. Steklova**213**(1997), no. Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 123–151 (Russian); English transl., Proc. Steklov Inst. Math.**2(213)**(1996), 115–140. MR**1632241** - Vadim Yu. Kaloshin,
*An extension of the Artin-Mazur theorem*, Ann. of Math. (2)**150**(1999), no. 2, 729–741. MR**1726706**, DOI 10.2307/121093 - Vadim Yu. Kaloshin,
*Generic diffeomorphisms with superexponential growth of number of periodic orbits*, Comm. Math. Phys.**211**(2000), no. 1, 253–271. MR**1757015**, DOI 10.1007/s002200050811 - 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. MR**1826992**, DOI 10.1090/S1079-6762-01-00090-7 - Robert Kaufman,
*On Hausdorff dimension of projections*, Mathematika**15**(1968), 153–155. MR**248779**, DOI 10.1112/S0025579300002503
Lieb1976 E. Lieb and W. Thirring, - 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**654892** - J. M. Marstrand,
*Some fundamental geometrical properties of plane sets of fractional dimensions*, Proc. London Math. Soc. (3)**4**(1954), 257–302. MR**63439**, DOI 10.1112/plms/s3-4.1.257 - 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** - Ian Melbourne and Ian Stewart,
*Symmetric $\omega$-limit sets for smooth $\Gamma$-equivariant dynamical systems with $\Gamma ^0$ abelian*, Nonlinearity**10**(1997), no. 6, 1551–1567. MR**1483554**, DOI 10.1088/0951-7715/10/6/007 - John Milnor,
*On the concept of attractor*, Comm. Math. Phys.**99**(1985), no. 2, 177–195. MR**790735** - T. Okon,
*Dimension estimate preserving embeddings for compacta in metric spaces*, Arch. Math. (Basel)**78**(2002), no. 1, 36–42. MR**1887314**, DOI 10.1007/s00013-002-8214-4
Ott2004 William Ott, Brian Hunt, and Vadim Kaloshin, - William Ott and James A. Yorke,
*Learning about reality from observation*, SIAM J. Appl. Dyn. Syst.**2**(2003), no. 3, 297–322. MR**2031277**, DOI 10.1137/S1111111102407421 - 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**97**, DOI 10.2307/1968940 - 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**, DOI 10.1215/S0012-7094-00-10222-0 - 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**, DOI 10.1007/978-94-010-0732-0 - Tim Sauer, James A. Yorke, and Martin Casdagli,
*Embedology*, J. Statist. Phys.**65**(1991), no. 3-4, 579–616. MR**1137425**, DOI 10.1007/BF01053745 - 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**, DOI 10.1017/S0143385797086252 - E. V. Shchepin and D. Repovš,
*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**, DOI 10.1007/BF02672712 - Barry Simon,
*Operators with singular continuous spectrum. I. General operators*, Ann. of Math. (2)**141**(1995), no. 1, 131–145. MR**1314033**, DOI 10.2307/2118629 - Masato Tsujii,
*Fat solenoidal attractors*, Nonlinearity**14**(2001), no. 5, 1011–1027. MR**1862809**, DOI 10.1088/0951-7715/14/5/306 - Hassler Whitney,
*Differentiable manifolds*, Ann. of Math. (2)**37**(1936), no. 3, 645–680. MR**1503303**, DOI 10.2307/1968482 - 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**, DOI 10.1023/A:1019762724717

*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. Birnir2001 —,

*Basic attractors and basic control of nonlinear partial differential equations*, Ecmi lecture notes, Chalmers University of Technology and Göteborg University, 2001.

*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.

*The effect of projections on fractal sets and measures in Banach spaces*, submitted to Ergodic Theory & Dynamical Systems, 2004.

## 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
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2149086