Where are typical C 1 functions one-to-one?

Zoltán Buczolich; András Máthé

Mathematica Bohemica (2006)

  • Volume: 131, Issue: 3, page 291-303
  • ISSN: 0862-7959

Abstract

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Suppose F ⊂ [ 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 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 α is the middle- α Cantor set we prove that the answer is yes if and only if dim ( C α ) ≤ 1 / 2 . There are F ’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.

How to cite

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Buczolich, Zoltán, and Máthé, András. "Where are typical $C^{1}$ functions one-to-one?." Mathematica Bohemica 131.3 (2006): 291-303. <http://eudml.org/doc/249909>.

@article{Buczolich2006,
abstract = {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.},
author = {Buczolich, Zoltán, Máthé, András},
journal = {Mathematica Bohemica},
keywords = {typical function; box dimension; one-to-one function; typical function; box dimension; one-to-one function},
language = {eng},
number = {3},
pages = {291-303},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Where are typical $C^\{1\}$ functions one-to-one?},
url = {http://eudml.org/doc/249909},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Buczolich, Zoltán
AU - Máthé, András
TI - Where are typical $C^{1}$ functions one-to-one?
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 3
SP - 291
EP - 303
AB - 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.
LA - eng
KW - typical function; box dimension; one-to-one function; typical function; box dimension; one-to-one function
UR - http://eudml.org/doc/249909
ER -

References

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  1. Dynamics of typical continuous functions, J. London Math. Soc. 40 (1989), 227–243. (1989) MR1044271
  2. 10.1016/j.aim.2004.11.009, Adv. Math. 201 (2006), 102–115. (2006) MR2204751DOI10.1016/j.aim.2004.11.009
  3. The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, 1985. (1985) Zbl0587.28004MR0867284
  4. Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, 1990. (1990) Zbl0689.28003MR1102677
  5. Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. (1995) Zbl0819.28004MR1333890
  6. Hausdorff Measures, Cambridge University Press, 1970. (1970) Zbl0204.37601MR0281862

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