| 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 |
| . |
