Local Dimensions and Quantization Dimensions in Dynamical Systems
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Local Dimensions and Quantization Dimensions in Dynamical Systems Mrinal Kanti Roychowdhury1 · Bilel Selmi2 Received: 25 July 2019 / Accepted: 7 October 2020 © Mathematica Josephina, Inc. 2020
Abstract Let μ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of Rk . We study the Hausdorff and the packing dimensions, and the quantization dimensions of μ with respect to the geometric mean error. The results establish the connections with various dimensions of the measure μ and generalize many known results about local dimensions and quantization dimensions of measures. Keywords Hyperbolic recurrent IFS · Irreducible row stochastic matrix · Local dimension · Hausdorff dimension · Packing dimension · Quantization dimension Mathematics Subject Classification Primary 37A50 · Secondary 28A80 · 94A34
1 Introduction Given a Borel probability measure μ on Rk , where k ∈ N, the nth quantization error for μ with respect to the geometric mean error is given by (1) en (μ) := inf exp log d(x, α)dμ(x) : α ⊂ Rk , 1 ≤ card(α) ≤ n , where d(x, α) denotes the distance between x and the set α with respect to an arbitrary norm d on Rk . A set α for which the infimum is achieved and contains no more than n
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Bilel Selmi [email protected] Mrinal Kanti Roychowdhury [email protected]
1
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999, USA
2
Analysis, Probability & Fractals Laboratory: LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000 Monastir, Tunisia
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M. K. Roychowdhury, B. Selmi
points is called an optimal set of n-means for μ, and the collection of all optimal sets of n-means for μ is denoted by Cn (μ). Under some suitable restriction, en (μ) tends to zero as n tends to infinity. Following [10], we write eˆn := eˆn (μ) = log en (μ) = inf
log d(x, α)dμ(x) : α ⊂ Rk , 1 ≤ card(α) ≤ n .
The numbers D(μ) := lim inf n→∞
log n log n and D(μ) := lim sup , −eˆn (μ) n→∞ −eˆn (μ)
are called the lower and the upper quantization dimensions of μ (of order zero), respectively. If D(μ) = D(μ), the common value is called the quantization dimension of μ and is denoted by D(μ). The quantization dimension measures the speed at which the specified measure of the error tends to zero as n tends to infinity. This problem arises in signal processing, data compression, cluster analysis, and pattern recognition, and it also has been studied in the context of economics, statistics, and numerical integration (see [3,7,11,14,20]). The quantization dimension with respect to the geometric mean error can be regarded as a limit state of that based on L r -metrics as r tends to zero (see [10, Lemma 3.5]). The following proposition gives a characterization of the lower and the upper quantization dimensions. Proposition 1.1 (see [10, Proposition 4.3]) Let D = D(μ) and D = D(μ). (a) If 0 ≤ t < D < s, then lim (log
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