PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2008 | 6 | 1 | 119-128
Tytuł artykułu

Transformations preserving the Hausdorff-Besicovitch dimension

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R 1 resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.
Wydawca
Czasopismo
Rocznik
Tom
6
Numer
1
Strony
119-128
Opis fizyczny
Daty
wydano
2008-03-01
online
2008-02-26
Bibliografia
  • [1] Albeverio S., Pratsiovytyi M., Torbin G., Fractal probability distributions and transformations preserving the Hausdorff-Besicovitch dimension, Ergodic Theory Dynam. Systems, 2004, 24, 1–16 http://dx.doi.org/10.1017/S0143385703000397
  • [2] Albeverio S., Torbin G., Fractal properties of singular continuous probability distributions with independent Q*-digits, Bull. Sci. Math., 2005, 129, 356–367 http://dx.doi.org/10.1016/j.bulsci.2004.12.001
  • [3] Billingsley P., Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965
  • [4] Billingsley P., Hausdorff dimension in probability theory II, Illinois J. Math., 1961, 5, 291–198
  • [5] Chatterji S.D., Certain induced measures and the fractional dimensions of their supports, Z. Wahrscheinlichkeits-theorie, 1964, 3, 184–192 http://dx.doi.org/10.1007/BF00534907
  • [6] Chatterji S.D., Certain induced measures on the unit interval, J. London Math. Soc., 1963, 38, 325–331 http://dx.doi.org/10.1112/jlms/s1-38.1.325
  • [7] Falconer K.J., Fractal geometry, John Wiley & Sons, Chichester, 1990
  • [8] Fractal Geometry and stochastics, Bandt Ch., Graf S., Zähle M. (Eds.), Birkhäuser Verlag, Basel, 2000
  • [9] Klein F., Verschiedene Betrachtung über neuere geometrische Forschunden, Erlangen, 1872
  • [10] Marsaglia G., Random variables with independent binary digits, Ann. Math. Statist., 1971, 42, 1922–1929 http://dx.doi.org/10.1214/aoms/1177693058
  • [11] Pratsiovytyi M.V., Fractal superfractal and anomalously fractal distribution of random variables with a fixed infinite set of independent n-adic digits, Exploring stochastic laws, VSP, 1995, 409–416
  • [12] Pratsiovytyi M.V., Fractal approach to investigations of singular probability distributions, National Pedagogical University, Kyiv, 1998
  • [13] Rogers C.A., Hausdorff measures, Cambridge University Press, Cambridge, 1998
  • [14] Salem R., On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 1943, 53, 423–439 http://dx.doi.org/10.2307/1990210
  • [15] Sauer T.D., Yorke J.A., Are the dimensions of a set and its image equal under typical smooth functions?, Ergodic Theory Dynam. Systems, 1997, 17, 941–956 http://dx.doi.org/10.1017/S0143385797086252
  • [16] Turbin A.F., Pratsiovytyi M.V., Fractal sets functions and distributions, Naukova Dumka, Kiev, 1992
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-008-0007-y
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.