Abstract
Let μ be a self-affine measure on a general Sierpiński carpet E. We give a characterization for the upper and lower quantization dimension of μ in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D r (μ) of μ exists for all r > 0 under an additional condition. We establish an explicit formula for D r (μ) and show that it increases to the box-counting dimension \({dim_B^* \mu}\) of μ as r tends to infinity. For a class of Sierpiński carpets E and the uniform measures μ on E, we show that the quantization dimension of μ coincides with its box-counting dimension and that the D r (μ)-dimensional upper and lower quantization coefficient of μ are both positive and finite.
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S. Zhu is supported by NNSF of China #10671150 and the Project-sponsored by SRF for ROCS, SEM.
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Zhu, S. The quantization dimension of the self-affine measures on general Sierpiński carpets. Monatsh Math 162, 355–374 (2011). https://doi.org/10.1007/s00605-009-0176-1
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DOI: https://doi.org/10.1007/s00605-009-0176-1
Keywords
- Quantization dimension
- Quantization coefficient
- Self-affine measure
- Sierpiński carpets
- Finite maximal anti-chain