Quantization dimensions of compactly supported probability measures via Rényi dimensions
نویسندگان
چکیده
We provide a complete picture of the upper quantization dimension in terms Rényi by proving that order r > 0 r>0 for an arbitrary compactly supported Borel probability measure alttext="nu"> ν encoding="application/x-tex">\nu is given its at point alttext="q Subscript r"> q encoding="application/x-tex">q_{r} where alttext="upper L Superscript q"> L encoding="application/x-tex">L^{q} -spectrum and line through origin with slope alttext="r"> encoding="application/x-tex">r intersect. In particular, this proves continuity right-arrow from bar D overbar r Baseline left-parenthesis reverse-solidus nu right-parenthesis"> ↦ D accent="false">¯<!-- ¯ </mml:mover> stretchy="false">( \nu ) encoding="application/x-tex">r\mapsto \overline {D}_{r}(\text {\nu )} as conjectured Lindsay [Quantization distributions, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University North Texas]. This viewpoint also sheds new light on connection problem other concepts fractal geometry we obtain one-to-one correspondence restricted to alttext="left-parenthesis 0 comma 1 ( , 1 ) encoding="application/x-tex">\left (0,1\right ) . give sufficient conditions existence dimension. way show byproduct exists every Gibbs respect alttext="script C 1"> C encoding="application/x-tex">\mathcal {C}^{1} -self-conformal iterated function system alttext="double-struck R d"> mathvariant="double-struck">R d encoding="application/x-tex">\mathbb {R}^{d} without any assumption separation well inhomogeneous self-similar measures under open sets condition. Some known general bounds dimensions can readily be derived our approach, some improved.
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2023
ISSN: ['2330-0000']
DOI: https://doi.org/10.1090/tran/8863