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2014 | 12 | 9 | 1305-1319
Tytuł artykułu

The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

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Very recently bounds for the L q spectra of inhomogeneous self-similar measures satisfying the Inhomogeneous Open Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities depending on positions. As an application of the results, we provide a systematic approach to obtaining non-trivial bounds for the L q spectra and Rényi dimension of inhomogeneous self-similar measures not satisfying the IOSC and of homogeneous ones not satisfying the OSC. We also provide some non-trivial bounds without any separation conditions.
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  • University of Silesia
  • [1] Arbeiter M., Patzschke N., Random self-similar multifractals, Math. Nachr., 1996, 181, 5–42
  • [2] Badii R., Politi A., Complexity, Cambridge Nonlinear Sci. Ser., 6, Cambridge University Press, Cambridge, 1997
  • [3] Bárány B., On the Hausdorff dimension of a family of self-similar sets with complicated overlaps, Fund. Math., 2009, 206, 49–59
  • [4] Barnsley M.F., Fractals Everywhere, 2nd ed., Academic Press, Boston, 1993
  • [5] Barnsley M.F., Superfractals, Cambridge University Press, Cambridge, 2006
  • [6] Barnsley M.F., Demko S., Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 1985, 399(1817), 243–275
  • [7] Beck C., Schlögl F., Thermodynamics of Chaotic Systems, Cambridge Nonlinear Sci. Ser., 4, Cambridge University Press, Cambridge, 1993
  • [8] Falconer K.J., Fractal Geometry, John Wiley & Sons, Chichester, 1990
  • [9] Falconer K., Techniques in Fractal Geometry, John Wiley & Sons, Chichester, 1997
  • [10] Feng D.-J., Gibbs properties of self-conformal measures and the multifractal formalism, Ergodic Theory Dynam. Systems, 2007, 27(3), 787–812
  • [11] Feng D.-J., Olivier E., Multifractal analysis of weak Gibbs measures and phase transition - application to some Bernoulli convolutions, Ergodic Theory Dynam. Systems, 2003, 23(6), 1751–1784
  • [12] Glickenstein D., Strichartz R.S., Nonlinear self-similar measures and their Fourier transforms, Indiana Univ. Math. J., 1996, 45(1), 205–220
  • [13] Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747
  • [14] Lasota A., Myjak J., On a dimension of measures, Bull. Polish Acad. Sci. Math., 2002, 50(2), 221–235
  • [15] Lau K.-S., Self-similarity, L p-spectrum and multifractal formalism, In: Fractal Geometry and Stochastics, Progr. Probab., 37, Birkhäuser, Basel, 1995, 55–90
  • [16] Lau K.-S., Ngai S.-M., Multifractal measures and a weak separation condition, Adv. Math., 1991, 141(1), 45–96
  • [17] Lau K.-S., Ngai S.-M., L q-spectrum of Bernoulli convolutions associated with P. V. numbers, Osaka J. Math., 1999, 36(4), 993–1010
  • [18] Liszka P., On inhomogeneous self-similar measures and their L q spectra, Ann. Polon. Math., 2013, 109(1), 75–92
  • [19] Olsen L., Snigireva N., L q spectra and Rényi dimensions of in-homogeneous self-similar measures, Nonlinearity, 2007, 20(1), 151–175
  • [20] Olsen L., Snigireva N., In-homogenous self-similar measures and their Fourier transforms, Math. Proc. Cambridge Philos. Soc., 2008, 144(2), 465–493
  • [21] Olsen L., Snigireva N., Multifractal spectra of in-homogenous self-similar measures, Indiana Univ. Math. J., 2008, 57(4), 1789–1844
  • [22] Peruggia M., Discrete Iterated Function Systems, AK Peters, Wellesley, MA, 1993
  • [23] Rényi A., Probability Theory, North-Holland Ser. Appl. Math. Mech., 10, North-Holland, Elsevier, Amsterdam-London, New York, 1970
  • [24] Testud B., Transitions de phase dans l’analyse multifractale de mesures auto-similaires, C. R. Math. Acad. Sci. Paris, 2005, 340(9), 653–658
  • [25] Testud B., Phase transitions for the multifractal analysis of self-similar measures, Nonlinearity, 2006, 19(5), 1201–1217
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