Geodesic
Where are typical $C^{1}$ functions one-to-one?
Mathematica Bohemica, Tome 131 (2006) no. 3, pp. 291-303

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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 ${\underline{\dim }}_{B}F1/2$ 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.
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 ${\underline{\dim }}_{B}F1/2$ 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.
DOI : 10.21136/MB.2006.134143
Classification : 26A15, 28A78, 28A80
Keywords: typical function; box dimension; one-to-one function 
Buczolich, Zoltán; Máthé, András. Where are typical $C^{1}$ functions one-to-one?. Mathematica Bohemica, Tome 131 (2006) no. 3, pp. 291-303. doi: 10.21136/MB.2006.134143
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