Quantization coefficients for ergodic measures on infinite symbolic space

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July 2014, 34(7): 2829-2846. doi: 10.3934/dcds.2014.34.2829

Quantization coefficients for ergodic measures on infinite symbolic space

Mrinal Kanti Roychowdhury ^1,

1.	Department of Mathematics,The University of Texas-Pan American, 1201 West University Drive, Edinburg, TX 78539-2999, United States

Received December 2012 Revised June 2013 Published December 2013

Abstract
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In this paper we consider an ergodic measure with bounded distortion on a symbolic space generated by an infinite alphabet, and showed that for each $r\in (0, +\infty)$ there exists a unique $k_r \in (0, +\infty)$ such that both the $k_r$-dimensional lower and upper quantization coefficients for its image measure $m$ with the support lying on the limit set generated by an infinite conformal iterated function system satisfying the strong open set condition are finite and positive. In addition, it shows that $k_r$ can be expressed by a simple formula involving the temperature function of the system. The result extends and generalizes a similar result of Roychowdhury established for a finite conformal iterated function system [Bull. Polish Acad. Sci. Math. 57 (2009)].

Keywords: Quantization coefficients, temperature function., ergodic measures with bounded distortion, quantization dimension, topological pressure.

Mathematics Subject Classification: Primary: 60Exx; Secondary: 37A50, 94A34, 28A8.

Citation: Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829

References:

[1]	K. J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1997.
[2]	K. J. Falconer, The multifractal spectrum of statistically self-similar measures, Journal of Theoretical Probability, 7 (1994), 681-701. doi: 10.1007/BF02213576.
[3]	A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academic Publishers, 1992. doi: 10.1007/978-1-4615-3626-0.
[4]	S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics 1730, Springer, Berlin, 2000. doi: 10.1007/BFb0103945.
[5]	S. Graf and H. Luschgy, The Quantization dimension of self-similar probabilities, Math. Nachr., 241 (2002), 103-109.
[6]	R. Gray and D. Neuhoff, Quantization, IEEE Trans. Inform. Theory, 44 (1998), 2325-2383. doi: 10.1109/18.720541.
[7]	J. Hutchinson, Fractals and self-similarity, Indiana Univ. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[8]	P. Hanus, R. D. Mauldin and M. Urbański, Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hung., 96 (2002), 27-98. doi: 10.1023/A:1015613628175.
[9]	L. J. Lindsay and R. D. Mauldin, Quantization dimension for conformal iterated function systems, Institute of Physics Publishing, Nonlinearity, 15 (2002), 189-199. doi: 10.1088/0951-7715/15/1/309.
[10]	R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154. doi: 10.1112/plms/s3-73.1.105.
[11]	R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543050.
[12]	N. Patzschke, Self-conformal multifractal measures, Adv. Appli. Math, 19 (1997), 486-513. doi: 10.1006/aama.1997.0557.
[13]	M. K. Roychowdhury, Lower quantization coefficient and the $F$-conformal measure, Colloquium Mathematicum, 122 (2011), 255-263. doi: 10.4064/cm122-2-11.
[14]	M. K. Roychowdhury, Quantization dimension function and ergodic measure with bounded distortion, Bulletin of the Polish Academy of Sciences Mathematics, 57 (2009), 251-262. doi: 10.4064/ba57-3-7.
[15]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.
[16]	K. Yoshida, Functional Analysis, Berlin-Heidelberg-New York: Springer, 1966.
[17]	S. Zhu, The lower quantization coefficient of the $F$-conformal measure is positive, Nonlinear Analysis, 69 (2008), 448-455. doi: 10.1016/j.na.2007.05.031.

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References:

[1]	K. J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1997.
[2]	K. J. Falconer, The multifractal spectrum of statistically self-similar measures, Journal of Theoretical Probability, 7 (1994), 681-701. doi: 10.1007/BF02213576.
[3]	A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academic Publishers, 1992. doi: 10.1007/978-1-4615-3626-0.
[4]	S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics 1730, Springer, Berlin, 2000. doi: 10.1007/BFb0103945.
[5]	S. Graf and H. Luschgy, The Quantization dimension of self-similar probabilities, Math. Nachr., 241 (2002), 103-109.
[6]	R. Gray and D. Neuhoff, Quantization, IEEE Trans. Inform. Theory, 44 (1998), 2325-2383. doi: 10.1109/18.720541.
[7]	J. Hutchinson, Fractals and self-similarity, Indiana Univ. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[8]	P. Hanus, R. D. Mauldin and M. Urbański, Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hung., 96 (2002), 27-98. doi: 10.1023/A:1015613628175.
[9]	L. J. Lindsay and R. D. Mauldin, Quantization dimension for conformal iterated function systems, Institute of Physics Publishing, Nonlinearity, 15 (2002), 189-199. doi: 10.1088/0951-7715/15/1/309.
[10]	R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154. doi: 10.1112/plms/s3-73.1.105.
[11]	R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, 148. Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543050.
[12]	N. Patzschke, Self-conformal multifractal measures, Adv. Appli. Math, 19 (1997), 486-513. doi: 10.1006/aama.1997.0557.
[13]	M. K. Roychowdhury, Lower quantization coefficient and the $F$-conformal measure, Colloquium Mathematicum, 122 (2011), 255-263. doi: 10.4064/cm122-2-11.
[14]	M. K. Roychowdhury, Quantization dimension function and ergodic measure with bounded distortion, Bulletin of the Polish Academy of Sciences Mathematics, 57 (2009), 251-262. doi: 10.4064/ba57-3-7.
[15]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.
[16]	K. Yoshida, Functional Analysis, Berlin-Heidelberg-New York: Springer, 1966.
[17]	S. Zhu, The lower quantization coefficient of the $F$-conformal measure is positive, Nonlinear Analysis, 69 (2008), 448-455. doi: 10.1016/j.na.2007.05.031.

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