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).

[2]

K. J. Falconer, The multifractal spectrum of statistically self-similar measures,, Journal of Theoretical Probability, 7 (1994), 681. 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, (1730). doi: 10.1007/BFb0103945.

[5]

S. Graf and H. Luschgy, The Quantization dimension of self-similar probabilities,, Math. Nachr., 241 (2002), 103.

[6]

R. Gray and D. Neuhoff, Quantization,, IEEE Trans. Inform. Theory, 44 (1998), 2325. doi: 10.1109/18.720541.

[7]

J. Hutchinson, Fractals and self-similarity,, Indiana Univ. J., 30 (1981), 713. 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. doi: 10.1023/A:1015613628175.

[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.

[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.

[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.

[12]

N. Patzschke, Self-conformal multifractal measures,, Adv. Appli. Math, 19 (1997), 486. doi: 10.1006/aama.1997.0557.

[13]

M. K. Roychowdhury, Lower quantization coefficient and the $F$-conformal measure,, Colloquium Mathematicum, 122 (2011), 255. 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. doi: 10.4064/ba57-3-7.

[15]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (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. doi: 10.1016/j.na.2007.05.031.

show all references

References:
[1]

K. J. Falconer, Techniques in Fractal Geometry,, John Wiley & Sons, (1997).

[2]

K. J. Falconer, The multifractal spectrum of statistically self-similar measures,, Journal of Theoretical Probability, 7 (1994), 681. 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, (1730). doi: 10.1007/BFb0103945.

[5]

S. Graf and H. Luschgy, The Quantization dimension of self-similar probabilities,, Math. Nachr., 241 (2002), 103.

[6]

R. Gray and D. Neuhoff, Quantization,, IEEE Trans. Inform. Theory, 44 (1998), 2325. doi: 10.1109/18.720541.

[7]

J. Hutchinson, Fractals and self-similarity,, Indiana Univ. J., 30 (1981), 713. 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. doi: 10.1023/A:1015613628175.

[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.

[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.

[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.

[12]

N. Patzschke, Self-conformal multifractal measures,, Adv. Appli. Math, 19 (1997), 486. doi: 10.1006/aama.1997.0557.

[13]

M. K. Roychowdhury, Lower quantization coefficient and the $F$-conformal measure,, Colloquium Mathematicum, 122 (2011), 255. 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. doi: 10.4064/ba57-3-7.

[15]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (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. doi: 10.1016/j.na.2007.05.031.

[1]

Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991

[2]

Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120.

[3]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[4]

Setsuro Fujiié, Jens Wittsten. Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3851-3873. doi: 10.3934/dcds.2018167

[5]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[6]

Mrinal Kanti Roychowdhury. Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4555-4570. doi: 10.3934/dcds.2018199

[7]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545

[8]

Luis Barreira, Christian Wolf. Dimension and ergodic decompositions for hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 201-212. doi: 10.3934/dcds.2007.17.201

[9]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018

[10]

Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118

[11]

De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699

[12]

M. Bulíček, Josef Málek, Dalibor Pražák. On the dimension of the attractor for a class of fluids with pressure dependent viscosities. Communications on Pure & Applied Analysis, 2005, 4 (4) : 805-822. doi: 10.3934/cpaa.2005.4.805

[13]

Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459

[14]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501

[15]

Huichi Huang. Fourier coefficients of $\times p$-invariant measures. Journal of Modern Dynamics, 2017, 11: 551-562. doi: 10.3934/jmd.2017021

[16]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237

[17]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[18]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779

[19]

Dou Dou. Minimal subshifts of arbitrary mean topological dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1411-1424. doi: 10.3934/dcds.2017058

[20]

Anatole Katok. Hyperbolic measures and commuting maps in low dimension. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 397-411. doi: 10.3934/dcds.1996.2.397

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]