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Quantization coefficients for ergodic measures on infinite symbolic space. (English) Zbl 1302.37007

Summary: In this paper, we consider an ergodic measure with bounded distortion on a symbolic space generated by an infinite alphabet and show that, for each \(r\in (0, +\infty)\), there exists a unique \(\kappa_r \in (0, +\infty)\) such that both the \(\kappa_r\)-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, we show that \(\kappa_r\) can be expressed by a simple formula involving the temperature function of the system. The result extends and generalizes a similar result established for a finite conformal iterated function system [the author, Bull. Pol. Acad. Sci., Math. 57, No. 3–4, 251–262 (2009; Zbl 1187.37015)].

MSC:

37A50 Dynamical systems and their relations with probability theory and stochastic processes
94A34 Rate-distortion theory in information and communication theory
28A80 Fractals

Citations:

Zbl 1187.37015
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Full Text: DOI

References:

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