The Hausdorff dimension of graphs of density continuous functions

Authors:
Zoltán Buczolich and Krzysztof Ostaszewski

Journal:
Proc. Amer. Math. Soc. **112** (1991), 1037-1043

MSC:
Primary 28A78; Secondary 26A15, 54C30

DOI:
https://doi.org/10.1090/S0002-9939-1991-1045588-X

MathSciNet review:
1045588

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Abstract: The density topology on the real line consists of all measurable sets whose all points are their points of Lebesgue density one. A real-valued function of a real variable that is continuous with respect to the density topology on both the domain and the range is termed *density continuous*. By constructing graphs of density continuous functions as invariant sets of systems of affine maps on the unit square we show that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two.

**[1]**M. Barnsley,*Fractals everywhere*, Academic Press, 1988. MR**977274 (90e:58080)****[2]**A. M. Bruckner,*Density-preserving homeomorphisms and the theorem of Maximoff*, Quart. J. Math. Oxford (2)**21**(1970), 337-347. MR**0268333 (42:3232)****[3]**M. Burke,*Some remarks on density-continuous functions*, Real Anal. Exchange**14**(1988-89), 235-242. MR**988370 (90a:26002)****[4]**K. Ciesielski and L. Larson,*The space of density continuous functions*, Acta Math. Hung, (to appear). MR**1153484 (92m:26004)****[5]**K. Ciesielski, L. Larson, and K. Ostaszewski,*Density continuity versus continuity*, Forum Math.**2**(1990), 265-275. MR**1050409 (91c:26003)****[6]**-,*Four continuities on the real line*, submitted.**[7]**-,*Differentiability and density continuity*, Real Anal. Exchange**15**(1) (1989-90), 239-248. MR**1042539 (91b:26007)****[8]**J. Malý,*The Peano curve and the density topology*, Real Anal. Exchange**5**(1979-80), 326-329. MR**574066 (81e:54037)****[9]**C. McMullen,*The Hausdorff dimension of general Sierpiński carpets*, Nagoya Math. J.**96**(1984), 1-9. MR**771063 (86h:11061)****[10]**J. Niewiarowski,*Density-preserving homeomorphisms*, Fund. Math.**106**(1980), 77-87. MR**580586 (81j:26012)****[11]**K. Ostaszewski,*Continuity in the density topology*, Real Anal. Exchange**7**(2) (1982), 259-270. MR**657324 (83e:26007)****[12]**-,*Continuity in the density topology*II, Rend. Circ. Mat. Palermo (2)**32**(1983), 398-414. MR**743202 (85g:26003)****[13]**J. C. Oxtoby,*Measure and category*, Springer-Verlag, New York, 1971. MR**584443 (81j:28003)****[14]**C. A. Rogers,*Hausdorff measures*, Cambridge University Press, 1970. MR**0281862 (43:7576)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1045588-X

Keywords:
Hausdorff dimension,
self-affine functions,
density topology,
density continuity

Article copyright:
© Copyright 1991
American Mathematical Society