Year: 2022
Type: article
Abstract: We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension of order r > 0 r>0 for an arbitrary compactly supported Borel probability measure ν \nu is given by its Rényi dimension at the point q r q_{r} where the L q L^{q} -spectrum of ν \nu and the line through the origin with slope r r intersect. In particular, this proves the continuity of r ↦ D ¯ r ( \nu ) r\mapsto \overline {D}_{r}(\text {\nu )} as conjectured by Lindsay [ Quantization dimension for probability distributions , ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of North Texas]. This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L q L^{q} -spectrum restricted to ( 0 , 1 ) \left (0,1\right ) . We give sufficient conditions in terms of the L q L^{q} -spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a C 1 \mathcal {C}^{1} -self-conformal iterated function system on R d
DOI: 10.1090/tran/8863
Authors: Marc Kesseböhmer, Aljoscha Niemann, Sanguo Zhu
Institutions: University of Bremen, Jiangsu University of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4281868153