Some dual statements concerning Wiener measure and Baire category
Author:
K. Simon
Journal:
Proc. Amer. Math. Soc. 106 (1989), 455463
MSC:
Primary 26A18; Secondary 26A27, 28C20
MathSciNet review:
961409
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Abstract: This paper deals with the duality between Wiener measure and Baire category on , the set of continuous functions endowed with the supremum norm. We prove that some properties shared by a residual set of continuous functions, e.g. the typical level set structure, and the existence of periodic points of order 3 hold a.e. with respect to the Wiener measure.
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 [1]
 S. J. Agronsky, A. M. Bruckner and M. Laczkovich, Dynamics of typical continuous functions (to appear in Journal of the London Mathematical Society). MR 1044271 (91e:26003)
 [2]
 A. M. Bruckner, Differentiation of real functions, Lect. Notes in Math. 659, SpringerVerlag, Berlin, 1978. MR 507448 (80h:26002)
 [3]
 W. A. Coppel, Iterates of continuous maps of an interval into itself, Lectures Notes in Math., 1984.
 [4]
 A. Dvoretsky, P. Erdös and S. Kakutani, Nonincreasing everywhere of the Brownian motion process, 4th Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability II, Univ. of California Press, 1961, pp. 103106.
 [5]
 HuiHsiung Kuo, Gaussian measures in banach spaces, Lect. Notes in Math. 463, SpringerVerlag, Berlin, 1975. MR 0461643 (57:1628)
 [6]
 K. Ito and H. P. McKean, Diffusion processes and their sample paths, SpringerVerlag, Berlin, 1965.
 [7]
 S. Karlin and H. M. Taylor, A first course in stochastic processes, Academic Press, New York, 1975. MR 0356197 (50:8668)
 [8]
 T. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly (1975) (82), 985992. MR 0385028 (52:5898)
 [9]
 S. Saks, Theory of the integral, Monograf. Mat. 7, WarsawsLwow, (1937).
 [10]
 S. Saks, On the functions of Besicovitch in the space of continuous functions, Fund. Math. 19 (1932), 211219.
 [11]
 K. Simon, Typical continuous functions are not iterates (to appear). MR 1077067 (91m:26005)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909614096
PII:
S 00029939(1989)09614096
Article copyright:
© Copyright 1989 American Mathematical Society
