# American Institute of Mathematical Sciences

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

## Quantization coefficients for ergodic measures on infinite symbolic space

 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

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)].
Citation: Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829
##### References:
 [1] K. J. Falconer, Techniques in Fractal Geometry,, John Wiley & Sons, (1997).   Google Scholar [2] K. J. Falconer, The multifractal spectrum of statistically self-similar measures,, Journal of Theoretical Probability, 7 (1994), 681.  doi: 10.1007/BF02213576.  Google Scholar [3] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression,, Kluwer Academic Publishers, (1992).  doi: 10.1007/978-1-4615-3626-0.  Google Scholar [4] S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions,, Lecture Notes in Mathematics 1730, (1730).  doi: 10.1007/BFb0103945.  Google Scholar [5] S. Graf and H. Luschgy, The Quantization dimension of self-similar probabilities,, Math. Nachr., 241 (2002), 103.   Google Scholar [6] R. Gray and D. Neuhoff, Quantization,, IEEE Trans. Inform. Theory, 44 (1998), 2325.  doi: 10.1109/18.720541.  Google Scholar [7] J. Hutchinson, Fractals and self-similarity,, Indiana Univ. J., 30 (1981), 713.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar [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.  doi: 10.1023/A:1015613628175.  Google Scholar [9] L. J. Lindsay and R. D. Mauldin, Quantization dimension for conformal iterated function systems,, Institute of Physics Publishing, 15 (2002), 189.  doi: 10.1088/0951-7715/15/1/309.  Google Scholar [10] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc., 73 (1996), 105.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar [11] R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,, Cambridge Tracts in Mathematics, (2003).  doi: 10.1017/CBO9780511543050.  Google Scholar [12] N. Patzschke, Self-conformal multifractal measures,, Adv. Appli. Math, 19 (1997), 486.  doi: 10.1006/aama.1997.0557.  Google Scholar [13] M. K. Roychowdhury, Lower quantization coefficient and the $F$-conformal measure,, Colloquium Mathematicum, 122 (2011), 255.  doi: 10.4064/cm122-2-11.  Google Scholar [14] M. K. Roychowdhury, Quantization dimension function and ergodic measure with bounded distortion,, Bulletin of the Polish Academy of Sciences Mathematics, 57 (2009), 251.  doi: 10.4064/ba57-3-7.  Google Scholar [15] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar [16] K. Yoshida, Functional Analysis,, Berlin-Heidelberg-New York: Springer, (1966).   Google Scholar [17] S. Zhu, The lower quantization coefficient of the $F$-conformal measure is positive,, Nonlinear Analysis, 69 (2008), 448.  doi: 10.1016/j.na.2007.05.031.  Google Scholar

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