# Article

 Title: Where are typical $C^{1}$ functions one-to-one? (English) Author: Buczolich, Zoltán Author: Máthé, András Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 131 Issue: 3 Year: 2006 Pages: 291-303 Summary lang: English . Category: math . 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. (English) Keyword: typical function Keyword: box dimension Keyword: one-to-one function MSC: 26A15 MSC: 28A78 MSC: 28A80 idZBL: Zbl 1112.26002 idMR: MR2248596 DOI: 10.21136/MB.2006.134143 . Date available: 2009-09-24T22:26:37Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134143 . Reference: [1] S. J. Agronsky, A. M. Bruckner, M. Laczkovich: Dynamics of typical continuous functions.J. London Math. Soc. 40 (1989), 227–243. MR 1044271 Reference: [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, 10.1016/j.aim.2004.11.009 Reference: [3] K. J. Falconer: The geometry of fractal sets.Cambridge Tracts in Mathematics, vol. 85, 1985. Zbl 0587.28004, MR 0867284 Reference: [4] K. J. Falconer: Fractal Geometry: Mathematical Foundations and Applications.John Wiley & Sons, 1990. Zbl 0689.28003, MR 1102677 Reference: [5] P. Mattila: Geometry of Sets and Measures in Euclidean Spaces.Cambridge University Press, 1995. Zbl 0819.28004, MR 1333890 Reference: [6] C. A. Rogers: Hausdorff Measures.Cambridge University Press, 1970. Zbl 0204.37601, MR 0281862 .

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