DIMENSIONALLY INVARIANT SPACES

  • Baek, In Soo (Department of Mathematics Pusan University of Foreign Studies)
  • Received : 2009.03.30
  • Accepted : 2009.05.15
  • Published : 2009.06.30

Abstract

We consider a code function from the unit interval which has a generalized dyadic expansion into a coding space which has an associated ultra metric. The code function is not a bi-Lipschitz map but a dimension-preserving map in the sense that the Hausdorff and packing dimensions of any subset in the unit interval and its image under the code function coincide respectively.

Keywords

References

  1. I. S. Baek, Relation between spectral classes of a self-similar Cantor set, J. Math. Anal. Appl. 292 (2004), no. 1, 294-302. https://doi.org/10.1016/j.jmaa.2003.12.001
  2. I. S. Baek, Multifractal analysis of a general coding space, J. Chungcheong Math. Soc. 19 (2006), no. 4, 357-364.
  3. I. S. Baek, L. Olsen and N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007), no. 1, 267-287. https://doi.org/10.1016/j.aim.2007.02.003
  4. I. S. Baek, Dimensions of distribution sets in the unit interval, Comm. Korean Math. Soc. 22 (2007), no. 4, 547-552. https://doi.org/10.4134/CKMS.2007.22.4.547
  5. I. S. Baek, Dimensionally equivalent spaces, J. Chungcheong Math. Soc. 21 (2008), no. 4, 527-532.
  6. C. D. Cutler, A note on equivalent interval covering systems for Hausdorff dimension on R, Intern. J. Math. & Math. Sci. 11 (1988), no. 4, 643-650. https://doi.org/10.1155/S016117128800078X
  7. Manav Das, Billingsley's packing dimension, Proc. Amer. Math. Soc. 136 (2008), no. 1, 273-278. https://doi.org/10.1090/S0002-9939-07-09069-7
  8. G. A. Edgar, Measure, Topology, and Fractal Geometry, Springer Verlag, 1990.
  9. K. J. Falconer, The Fractal Geometry, John Wiley & Sons , 1990.
  10. H. H. Lee and I. S. Baek, A note on equivalent interval covering systems for packing dimension of R, J. Korean Math. Soc. 28 (1991), 195-205.
  11. L. Olsen, Extremely non-normal numbers, Math. Proc. Camb. Phil. Soc. 137 (2004), 43-53. https://doi.org/10.1017/S0305004104007601