Quantization dimension for some Moran measures
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- by Mrinal Kanti Roychowdhury PDF
- Proc. Amer. Math. Soc. 138 (2010), 4045-4057 Request permission
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
- Robert Cawley and R. Daniel Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), no. 2, 196–236. MR 1155465, DOI 10.1016/0001-8708(92)90064-R
- Meifeng Dai and Ying Jiang, Fractal dimension and measure of the subset of Moran set, Chaos Solitons Fractals 40 (2009), no. 1, 190–196. MR 2517925, DOI 10.1016/j.chaos.2007.07.042
- Meifeng Dai and Dehua Liu, The local dimension of Moran measures satisfying the strong separation condition, Chaos Solitons Fractals 38 (2008), no. 4, 1025–1030. MR 2435601, DOI 10.1016/j.chaos.2007.02.013
- M. Dai and X. Tan, Quantization dimension of random self-similar measures, J. Math. Anal. Appl., 362 (2010), 471-475.
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
- K. J. Falconer, The multifractal spectrum of statistically self-similar measures, J. Theoret. Probab. 7 (1994), no. 3, 681–702. MR 1284660, DOI 10.1007/BF02213576
- A. Gersho and R. M. Gray, Vector quantization and signal compression, Kluwer Academic Publishers: Boston, 1992.
- Siegfried Graf and Harald Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics, vol. 1730, Springer-Verlag, Berlin, 2000. MR 1764176, DOI 10.1007/BFb0103945
- S. Graf and H. Luschgy, Asymptotics of the quantization errors for self-similar probabilities, Real Anal. Exchange 26 (2000/01), no. 2, 795–810. MR 1844394
- Siegfried Graf and Harald Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr. 241 (2002), 103–109. MR 1912380, DOI 10.1002/1522-2616(200207)241:1<103::AID-MANA103>3.0.CO;2-J
- Thomas C. Halsey, Mogens H. Jensen, Leo P. Kadanoff, Itamar Procaccia, and Boris I. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A (3) 33 (1986), no. 2, 1141–1151. MR 823474, DOI 10.1103/PhysRevA.33.1141
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Su Hua, Hui Rao, Zhiying Wen, and Jun Wu, On the structures and dimensions of Moran sets, Sci. China Ser. A 43 (2000), no. 8, 836–852. MR 1799919, DOI 10.1007/BF02884183
- Antti Käenmäki and Markku Vilppolainen, Separation conditions on controlled Moran constructions, Fund. Math. 200 (2008), no. 1, 69–100. MR 2415480, DOI 10.4064/fm200-1-2
- Wolfgang Kreitmeier, Optimal quantization for dyadic homogeneous Cantor distributions, Math. Nachr. 281 (2008), no. 9, 1307–1327. MR 2442708, DOI 10.1002/mana.200510680
- L. J. Lindsay and R. D. Mauldin, Quantization dimension for conformal iterated function systems, Nonlinearity 15 (2002), no. 1, 189–199. MR 1877974, DOI 10.1088/0951-7715/15/1/309
- Min Wu, The multifractal spectrum of some Moran measures, Sci. China Ser. A 48 (2005), no. 8, 1097–1112. MR 2180104, DOI 10.1360/022004-10
- Norbert Patzschke, Self-conformal multifractal measures, Adv. in Appl. Math. 19 (1997), no. 4, 486–513. MR 1479016, DOI 10.1006/aama.1997.0557
- Yuval Peres, MichałRams, Károly Simon, and Boris Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc. 129 (2001), no. 9, 2689–2699. MR 1838793, DOI 10.1090/S0002-9939-01-05969-X
- M. K. Roychowdhury, Quantization dimension function and ergodic measure with bounded distortion, Bulletin of the Polish Academy of Sciences, 57 (2009), 251-262.
- Andreas Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), no. 1, 111–115. MR 1191872, DOI 10.1090/S0002-9939-1994-1191872-1
- P. L. Zador, Development and evaluation of procedures for quantizing multivariate distributions, PhD thesis, Stanford University (1964).
- Sanguo Zhu, Quantization dimension of probability measures supported on Cantor-like sets, J. Math. Anal. Appl. 338 (2008), no. 1, 742–750. MR 2386455, DOI 10.1016/j.jmaa.2007.05.004
- S. Zhu, The quantization dimension of the self-affine measures on general Sierpi$\acute {n}$ski carpets, Monatshefte für Mathematik, published electronically, 27 November 2009.
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
- Received by editor(s): November 12, 2009
- Received by editor(s) in revised form: January 22, 2010
- Published electronically: May 17, 2010
- Communicated by: Bryna Kra
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 4045-4057
- MSC (2010): Primary 37A50; Secondary 28A80, 94A34
- DOI: https://doi.org/10.1090/S0002-9939-2010-10406-9
- MathSciNet review: 2679625