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

Przemysław Liszka

Open Mathematics (2014)

  • Volume: 12, Issue: 9, page 1305-1319
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Przemysław Liszka. "The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures." Open Mathematics 12.9 (2014): 1305-1319. <http://eudml.org/doc/269471>.

@article{PrzemysławLiszka2014,
abstract = {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.},
author = {Przemysław Liszka},
journal = {Open Mathematics},
keywords = {Lq spectra; Rényi dimension; Inhomogeneous self-similar measure; Homogeneous self-similar measure; spectra; inhomogeneous self-similar measure; homogeneous self-similar measure},
language = {eng},
number = {9},
pages = {1305-1319},
title = {The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures},
url = {http://eudml.org/doc/269471},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Przemysław Liszka
TI - The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures
JO - Open Mathematics
PY - 2014
VL - 12
IS - 9
SP - 1305
EP - 1319
AB - 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.
LA - eng
KW - Lq spectra; Rényi dimension; Inhomogeneous self-similar measure; Homogeneous self-similar measure; spectra; inhomogeneous self-similar measure; homogeneous self-similar measure
UR - http://eudml.org/doc/269471
ER -

References

top
  1. [1] Arbeiter M., Patzschke N., Random self-similar multifractals, Math. Nachr., 1996, 181, 5–42 http://dx.doi.org/10.1002/mana.3211810102 Zbl0873.28003
  2. [2] Badii R., Politi A., Complexity, Cambridge Nonlinear Sci. Ser., 6, Cambridge University Press, Cambridge, 1997 http://dx.doi.org/10.1017/CBO9780511524691 Zbl1042.82500
  3. [3] Bárány B., On the Hausdorff dimension of a family of self-similar sets with complicated overlaps, Fund. Math., 2009, 206, 49–59 http://dx.doi.org/10.4064/fm206-0-4 Zbl1194.28006
  4. [4] Barnsley M.F., Fractals Everywhere, 2nd ed., Academic Press, Boston, 1993 
  5. [5] Barnsley M.F., Superfractals, Cambridge University Press, Cambridge, 2006 http://dx.doi.org/10.1017/CBO9781107590168 
  6. [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 http://dx.doi.org/10.1098/rspa.1985.0057 Zbl0588.28002
  7. [7] Beck C., Schlögl F., Thermodynamics of Chaotic Systems, Cambridge Nonlinear Sci. Ser., 4, Cambridge University Press, Cambridge, 1993 http://dx.doi.org/10.1017/CBO9780511524585 
  8. [8] Falconer K.J., Fractal Geometry, John Wiley & Sons, Chichester, 1990 
  9. [9] Falconer K., Techniques in Fractal Geometry, John Wiley & Sons, Chichester, 1997 Zbl0869.28003
  10. [10] Feng D.-J., Gibbs properties of self-conformal measures and the multifractal formalism, Ergodic Theory Dynam. Systems, 2007, 27(3), 787–812 http://dx.doi.org/10.1017/S0143385706000952 Zbl1126.28003
  11. [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 http://dx.doi.org/10.1017/S0143385703000051 Zbl1128.37303
  12. [12] Glickenstein D., Strichartz R.S., Nonlinear self-similar measures and their Fourier transforms, Indiana Univ. Math. J., 1996, 45(1), 205–220 http://dx.doi.org/10.1512/iumj.1996.45.1156 Zbl0860.28007
  13. [13] Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747 http://dx.doi.org/10.1512/iumj.1981.30.30055 Zbl0598.28011
  14. [14] Lasota A., Myjak J., On a dimension of measures, Bull. Polish Acad. Sci. Math., 2002, 50(2), 221–235 Zbl1020.28004
  15. [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 http://dx.doi.org/10.1007/978-3-0348-7755-8_4 
  16. [16] Lau K.-S., Ngai S.-M., Multifractal measures and a weak separation condition, Adv. Math., 1991, 141(1), 45–96 http://dx.doi.org/10.1006/aima.1998.1773 Zbl0929.28007
  17. [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 Zbl0956.28010
  18. [18] Liszka P., On inhomogeneous self-similar measures and their L q spectra, Ann. Polon. Math., 2013, 109(1), 75–92 http://dx.doi.org/10.4064/ap109-1-6 Zbl06176235
  19. [19] Olsen L., Snigireva N., L q spectra and Rényi dimensions of in-homogeneous self-similar measures, Nonlinearity, 2007, 20(1), 151–175 http://dx.doi.org/10.1088/0951-7715/20/1/010 Zbl1124.28010
  20. [20] Olsen L., Snigireva N., In-homogenous self-similar measures and their Fourier transforms, Math. Proc. Cambridge Philos. Soc., 2008, 144(2), 465–493 http://dx.doi.org/10.1017/S0305004107000771 Zbl1146.28005
  21. [21] Olsen L., Snigireva N., Multifractal spectra of in-homogenous self-similar measures, Indiana Univ. Math. J., 2008, 57(4), 1789–1844 http://dx.doi.org/10.1512/iumj.2008.57.3622 Zbl1218.28005
  22. [22] Peruggia M., Discrete Iterated Function Systems, AK Peters, Wellesley, MA, 1993 Zbl0788.60086
  23. [23] Rényi A., Probability Theory, North-Holland Ser. Appl. Math. Mech., 10, North-Holland, Elsevier, Amsterdam-London, New York, 1970 
  24. [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 http://dx.doi.org/10.1016/j.crma.2005.03.020 Zbl1077.28009
  25. [25] Testud B., Phase transitions for the multifractal analysis of self-similar measures, Nonlinearity, 2006, 19(5), 1201–1217 http://dx.doi.org/10.1088/0951-7715/19/5/009 Zbl1093.28007

NotesEmbed ?

top

You must be logged in to post comments.