Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system

[2503.11105] Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2503.11105 (math) [Submitted on 14 Mar 2025 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: Quantization dimension for a generalized inhomogeneous bi-Lipschitz iterated function system Authors: Shivam Dubey , Mrinal Kanti Roychowdhury , Saurabh Verma View PDF HTML (experimental) Abstract: For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(\mu)$, of a Borel probability measure $\mu$ 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 $\mu$ tends to infinity. If $D_r(\mu)$ does not exists, we call $\underline D_r(\mu)$ and $\overline D_r(\mu)$, the lower and upper quantization dimensions of $\mu$ of order $r$. In this paper, we estimate the quantization dimension of condensation measures associated with condensation systems $(\{f_i\}_{i=1}^N, (p_i)_{i=0}^N, \nu)$, where the mappings $f_i$ are bi-Lipschitz and the measure $\nu$ 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. Subjects: Dynamical Systems (math.DS) ; Probability (math.PR) MSC classes: 60Exx, 94A34, 28A80 Cite as: arXiv:2503.11105 [math.DS] (or arXiv:2503.11105v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2503.11105 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mrinal Kanti Roychowdhury [ view email ] [v1] Fri, 14 Mar 2025 05:52:16 UTC (17 KB) [v2] Mon, 24 Aug 2026 20:47:08 UTC (18 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2025-03 Change to browse by: math math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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