Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces
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- by Brian R. Hunt, Tim Sauer and James A. Yorke PDF
- Bull. Amer. Math. Soc. 27 (1992), 217-238 Request permission
Abstract:
We present a measure-theoretic condition for a property to hold "almost everywhere" on an infinite-dimensional vector space, with particular emphasis on function spaces such as ${C^k}$ and ${L^p}$. Like the concept of "Lebesgue almost every" on finite-dimensional spaces, our notion of "prevalence" is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of "open and dense" or "generic" when one desires a probabilistic result on the likelihood of a given property on a function space. We give several examples of properties which hold "almost everywhere" in the sense of prevalence. For instance, we prove that almost every ${C^1}$ map on ${\mathbb {R}^n}$ has the property that all of its periodic orbits are hyperbolic.References
- V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 250, Springer-Verlag, New York-Berlin, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR 695786, DOI 10.1007/978-1-4684-0147-9
- Paul Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 85–141. MR 741725, DOI 10.1090/S0273-0979-1984-15240-6
- Hubert Cremer, Zum Zentrumproblem, Math. Ann. 98 (1928), no. 1, 151–163 (German). MR 1512397, DOI 10.1007/BF01451586 N. Dunford and J. T. Schwartz, Linear operators, Part 1, Interscience, New York, 1958. I. V. Girsanov and B. S. Mityagin, Quasi-invariant measures and linear topological spaces, Nauchn. Dokl. Vys. Skol. 2 (1959), 5-10. (Russian)
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
- Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0348781
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
- 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
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
- Judy A. Kennedy and James A. Yorke, Pseudocircles in dynamical systems, Trans. Amer. Math. Soc. 343 (1994), no. 1, 349–366. MR 1187029, DOI 10.1090/S0002-9947-1994-1187029-5
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- M. Yu. Lyubich, Generic behavior of trajectories of the exponential function, Uspekhi Mat. Nauk 41 (1986), no. 2(248), 199–200 (Russian). MR 842176
- M. Yu. Lyubich, The measurable dynamics of the exponential, Sibirsk. Mat. Zh. 28 (1987), no. 5, 111–127 (Russian). MR 924986
- J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale. MR 0494309, DOI 10.1007/978-1-4612-6374-6
- MichałMisiurewicz, On iterates of $e^{z}$, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 103–106. MR 627790, DOI 10.1017/s014338570000119x J. Mycielski, Unsolved problems on the prevalence of ergodicity, instability and algebraic independence, The Ulam Quarterly (to appear).
- Sheldon E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101–151. MR 556584, DOI 10.1007/BF02684771
- Helena E. Nusse and Laura Tedeschini-Lalli, Wild hyperbolic sets, yet no chance for the coexistence of infinitely many KLUS-simple Newhouse attracting sets, Comm. Math. Phys. 144 (1992), no. 3, 429–442. MR 1158755, DOI 10.1007/BF02099177
- John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443, DOI 10.1007/978-1-4684-9339-9
- 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
- Charles Pugh and Michael Shub, Ergodicity of Anosov actions, Invent. Math. 15 (1972), 1–23. MR 295390, DOI 10.1007/BF01418639
- Frank Quinn and Arthur Sard, Hausdorff conullity of critical images of Fredholm maps, Amer. J. Math. 94 (1972), 1101–1110. MR 322899, DOI 10.2307/2373565
- Mary Rees, The exponential map is not recurrent, Math. Z. 191 (1986), no. 4, 593–598. MR 832817, DOI 10.1007/BF01162349
- Clark Robinson, Bifurcation to infinitely many sinks, Comm. Math. Phys. 90 (1983), no. 3, 433–459. MR 719300, DOI 10.1007/BF01206892
- Arthur Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883–890. MR 7523, DOI 10.1090/S0002-9904-1942-07811-6 T. Sauer and J. A. Yorke, Statistically self-similar sets, preprint.
- 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
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- Carl Ludwig Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612. MR 7044, DOI 10.2307/1968952
- V. N. Sudakov, Linear sets with quasi-invariant measure, Dokl. Akad. Nauk SSSR 127 (1959), 524–525 (Russian). MR 0107689
- V. N. Sudakov, On quasi-invariant measures in linear spaces, Vestnik Leningrad. Univ. 15 (1960), no. 19, 5–8 (Russian, with English summary). MR 0152627
- Laura Tedeschini-Lalli and James A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys. 106 (1986), no. 4, 635–657. MR 860314, DOI 10.1007/BF01463400
- Hassler Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645–680. MR 1503303, DOI 10.2307/1968482
- Y. Yamasaki, Measures on infinite-dimensional spaces, Series in Pure Mathematics, vol. 5, World Scientific Publishing Co., Singapore, 1985. MR 999137, DOI 10.1142/0162
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 27 (1992), 217-238
- MSC (2000): Primary 28C20; Secondary 46G12
- DOI: https://doi.org/10.1090/S0273-0979-1992-00328-2
- MathSciNet review: 1161274