Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system
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| Format: | Preprint |
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2025
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| _version_ | 1866929759769853952 |
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| author | Dubey, Shivam Roychowdhury, Mrinal Kanti Verma, Saurabh |
| author_facet | Dubey, Shivam Roychowdhury, Mrinal Kanti Verma, Saurabh |
| contents | For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on ${\mathbb R}^d$ represents the speed how fast the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exists, we call $\underline D_r(μ)$ and $\overline D_r(μ)$, the lower and upper quantization dimensions of $μ$ of order $r$. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems $(\{f_i\}_{i=1}^N, (p_i)_{i=0}^N, ν)$, where the mappings $f_i$ are bi-Lipschitz and the measure $ν$ is an image measure of an ergodic measure with bounded distortion supported on a conformal set. In addition, we determine the optimal quantization for an infinite discrete distribution, and give an example which shows that the quantization dimension of a Borel probability measure can be positive with zero quantization coefficient. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_11105 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system Dubey, Shivam Roychowdhury, Mrinal Kanti Verma, Saurabh Dynamical Systems Probability 60Exx, 94A34, 28A80 For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on ${\mathbb R}^d$ represents the speed how fast the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. If $D_r(μ)$ does not exists, we call $\underline D_r(μ)$ and $\overline D_r(μ)$, the lower and upper quantization dimensions of $μ$ of order $r$. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems $(\{f_i\}_{i=1}^N, (p_i)_{i=0}^N, ν)$, where the mappings $f_i$ are bi-Lipschitz and the measure $ν$ is an image measure of an ergodic measure with bounded distortion supported on a conformal set. In addition, we determine the optimal quantization for an infinite discrete distribution, and give an example which shows that the quantization dimension of a Borel probability measure can be positive with zero quantization coefficient. |
| title | Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system |
| topic | Dynamical Systems Probability 60Exx, 94A34, 28A80 |
| url | https://arxiv.org/abs/2503.11105 |