# Article

Full entry | PDF   (0.3 MB)
Keywords:
typical continuous function; Brownian motion; Takagi’s function; Weierstrass’s function
Summary:
Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set \$A\$ we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s function is graph like, and Weierstrass’s nowhere differentiable function is central graph like.
References:
[1] P. Billingsley: Probability and Measure. Third edition. John Wiley, Chichester, 1995. MR 1324786
[2] K. Falconer: Fractal Geometry. John Wiley, Chichester, 1990. MR 1102677 | Zbl 0689.28003
[3] K. Falconer: Techniques in Fractal Geometry. John Wiley, Chichester, 1997. MR 1449135 | Zbl 0869.28003
[4] K. Falconer: Tangent fields and the local structure of random fields. J. Theoret. Probab. 15 (2002), no. 3, 731–750. DOI 10.1023/A:1016276016983 | MR 1922445 | Zbl 1013.60028
[5] K. Falconer: The local structure of random processes. Preprint. MR 1967698 | Zbl 1054.28003
[6] H. Fürstenberg: Ergodic Theory and the Geometry of Fractals, talk given at the conference Fractals in Graz, 2001, http://finanz.math.tu-graz.ac.at/\$\sim \$fractal.
[7] B. R. Gelbaum: Modern Real and Complex Analysis. John Wiley, New York, 1995. MR 1325692
[8] G. H. Hardy: Weierstrass’s non-differentiable function. Trans. Amer. Math. Soc. 17 (1916), 301–325. MR 1501044
[9] P. Humke, G. Petruska: The packing dimension of a typical continuous function is 2. Real Anal. Exch. 14 (1988–89), 345–358. MR 0995975
[10] S. Jaffard: Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3 (1997), 1–22. DOI 10.1007/BF02647944 | MR 1428813 | Zbl 0880.28007
[11] S. V. Levizov: On the central limit theorem for series with respect to periodical multiplicative systems I. Acta Sci. Math. (Szeged) 55 (1991), 333–359. MR 1152596 | Zbl 0759.42018
[12] S. V. Levizov: Weakly lacunary trigonometric series. Izv. Vyssh. Uchebn. Zaved. Mat. (1988), 28–35, 86–87. MR 0938430 | Zbl 0713.42011
[13] N. N. Luzin: Sur les propriétés des fonctions mesurables. C. R. Acad. Sci. Paris 154 (1912), 1688–1690.
[14] P. Mattila: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, 1995. MR 1333890 | Zbl 0819.28004
[15] P. Mattila: Tangent measures, densities, and singular integrals. Fractal geometry and stochastics (Finsterbergen, 1994), 43–52, Progr. Probab. 37, Birkhäuser, Basel, 1995. MR 1391970 | Zbl 0837.28006
[16] R. D. Mauldin, S. C. Williams: On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298 (1986), 793–803. DOI 10.1090/S0002-9947-1986-0860394-7 | MR 0860394
[17] D. Preiss: Geometry of measures in \$\mathbb{R}^{n}\$: distribution, rectifiability, and densities. Ann. Math., II. Ser. 125 (1987), 537–643. DOI 10.2307/1971410 | MR 0890162
[18] D. Preiss, L. Zajíček: On Dini and approximate Dini derivates of typical continuous functions. Real Anal. Exch. 26 (2000/01), 401–412. MR 1825518
[19] S. Saks: Theory of the Integral. Second Revised (ed.), Dover, New York, 1964. MR 0167578
[20] L. Zajíček: On preponderant differentiability of typical continuous functions. Proc. Amer. Math. Soc. 124 (1996), 789–798. DOI 10.1090/S0002-9939-96-03057-2 | MR 1291796

Partner of