|
Quantization dimension for some Moran measures
Author(s):
Mrinal
Kanti
Roychowdhury
Journal:
Proc. Amer. Math. Soc.
138
(2010),
4045-4057.
MSC (2010):
Primary 37A50;
Secondary 28A80, 94A34
Posted:
May 17, 2010
MathSciNet review:
2679625
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
The quantization dimension function for some Moran measures has been determined, and a relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is established.
References:
-
- 1.
- R. Cawley and R. D. Mauldin, Multifractal decomposition of Moran fractals, Advances in Mathematics, 92 (1992), 196-236. MR 1155465 (93b:58085)
- 2.
- M. Dai and Y. Jiang, Fractal dimension and measure of the subset of Moran set, Chaos, Solitons and Fractals, 40 (2009), 190-196. MR 2517925
- 3.
- M. Dai and D. Liu, The local dimension of Moran measures satisfying the strong separation condition, Chaos, Solitons and Fractals, 38 (2008), 1025-1030. MR 2435601 (2009g:28015)
- 4.
- M. Dai and X. Tan, Quantization dimension of random self-similar measures, J. Math. Anal. Appl., 362 (2010), 471-475.
- 5.
- K. J. Falconer, Techniques in fractal geometry, Chichester: Wiley, 1997. MR 1449135 (99f:28013)
- 6.
- K. J. Falconer, The multifractal spectrum of statistically self-similar measures, Journal of Theoretical Probability, Vol. 7, No. 3, 681-702, 1994. MR 1284660 (95m:60076)
- 7.
- A. Gersho and R. M. Gray, Vector quantization and signal compression, Kluwer Academic Publishers: Boston, 1992.
- 8.
- S. Graf and H. Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics, 1730, Springer, Berlin, 2000. MR 1764176 (2001m:60043)
- 9.
- S. Graf and H. Luschgy, Asymptotics of the quantization errors for self-similar probabilities, Real Anal. Exchange, 26, 795-810, 2001. MR 1844394 (2002g:28011)
- 10.
- S. Graf and H. Luschgy, The Quantization Dimension of Self-Similar Probabilities, Math. Nachr., 241 (2002), 103-109. MR 1912380 (2003j:28025)
- 11.
- T. Halsey, M. Jensen, L. Kadanoff and I. Procaccia, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A (3), 33, 1141-51, 1986. MR 823474 (87h:58125a)
- 12.
- J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30, 713-747, 1981. MR 625600 (82h:49026)
- 13.
- S. Hua, H. Rao, Z. Wen, J. Wu, On the structures and dimensions of Moran sets, Science of China (Series A), 43 (8) (2000), 836-852. MR 1799919 (2002a:28011)
- 14.
- A. Käenmäki and M. Vilppolainen, Separation conditions on controlled Moran constructions, Fund. Math., 200 (2008), 69-100. MR 2415480 (2010a:28010)
- 15.
- W. Kreitmeier, Optimal quantization for dyadic homogeneous Cantor distributions, Math. Nachr., 281, No. 9, 1307-1327 (2008). MR 2442708 (2010c:94025)
- 16.
- L. J. Lindsay and R. D. Mauldin, Quantization dimension for conformal iterated function systems, Institute of Physics Publishing, Nonlinearity, 15 (2002), 189-199. MR 1877974 (2002j:28014)
- 17.
- W. Min, The multifractal spectrum of some Moran measures, Science in China Ser. A Mathematics, 2005, Vol. 48, No. 8, 1097-1112. MR 2180104 (2006j:28022)
- 18.
- N. Patzschke, Self-conformal multifractal measures, Adv. in Appl. Math., Volume 19, Issue 4 (1997), 486-513. MR 1479016 (99c:28020)
- 19.
- Y. Peres, M. Rams, K. Simon and B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc., 129 (2001), 2689-2699. MR 1838793 (2002d:28004)
- 20.
- M. K. Roychowdhury, Quantization dimension function and ergodic measure with bounded distortion, Bulletin of the Polish Academy of Sciences, 57 (2009), 251-262.
- 21.
- A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115. MR 1191872 (94k:28012)
- 22.
- P. L. Zador, Development and evaluation of procedures for quantizing multivariate distributions, PhD thesis, Stanford University (1964).
- 23.
- S. Zhu, Quantization dimension of probability measures supported on Cantor-like sets, J. Math. Anal. Appl., vol. 338, no. 1, 2008, 742-750. MR 2386455 (2009c:28025)
- 24.
- S. Zhu, The quantization dimension of the self-affine measures on general Sierpi
ski carpets, Monatshefte für Mathematik, published electronically, 27 November 2009.
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2010):
37A50,
28A80, 94A34
Retrieve articles in all Journals with
MSC (2010):
37A50,
28A80, 94A34
Additional Information:
Mrinal
Kanti
Roychowdhury
Affiliation:
Department of Mathematics, The University of Texas-Pan American, 1201 West University Drive, Edinburg, Texas 78539-2999
Email:
roychowdhurymk@utpa.edu
DOI:
10.1090/S0002-9939-2010-10406-9
PII:
S 0002-9939(2010)10406-9
Keywords:
Moran measure,
quantization dimension,
temperature function
Received by editor(s):
November 12, 2009
Received by editor(s) in revised form:
January 22, 2010
Posted:
May 17, 2010
Communicated by:
Bryna Kra
Copyright of article:
Copyright
2010,
American Mathematical Society
|