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Keywords:
typical function; box dimension; one-to-one function
Summary:
Suppose $F\subset [0,1]$ is closed. Is it true that the typical (in the sense of Baire category) function in $C^{1}[0,1]$ is one-to-one on $F$? If ${\underline{\dim }}_{B}F<1/2$ we show that the answer to this question is yes, though we construct an $F$ with $\dim _{B}F=1/2$ for which the answer is no. If $C_{\alpha }$ is the middle-$\alpha $ Cantor set we prove that the answer is yes if and only if $\dim (C_{\alpha })\le 1/2.$ There are $F$’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.
References:
[1] S. J. Agronsky, A. M. Bruckner, M. Laczkovich: Dynamics of typical continuous functions. J. London Math. Soc. 40 (1989), 227–243. MR 1044271
[2] M. Elekes, T. Keleti: Borel sets which are null or non-sigma-finite for every translation invariant measure. Adv. Math. 201 (2006), 102–115. MR 2204751
[3] K. J. Falconer: The geometry of fractal sets. Cambridge Tracts in Mathematics, vol. 85, 1985. MR 0867284 | Zbl 0587.28004
[4] K. J. Falconer: Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, 1990. MR 1102677 | Zbl 0689.28003
[5] P. Mattila: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, 1995. MR 1333890 | Zbl 0819.28004
[6] C. A. Rogers: Hausdorff Measures. Cambridge University Press, 1970. MR 0281862 | Zbl 0204.37601
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