Geodesic
On the Intermediate Values
Matematičeskie zametki, Tome 115 (2024) no. 3, pp. 348-354 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is well known that the lower dimension $\underline{D}(\mu)$ of quantization of a Borel probability measure $\mu$ given on a metric compactum $(X,\rho)$ does not exceed the lower capacitive dimension $\underline{\dim}_BX$ of $X$. We prove the following theorem on the intermediate values of the lower dimension of quantization of probability measures: for any nonnegative number $a$ smaller that the dimension $z\underline{\dim}_BX$ of the compactum $X$, there exists a probability measure $\mu_a$ on $X$ with support $X$ such that $\underline{D}(\mu_a)=a$. The number $z\underline{\dim}_BX$ characterizes the asymptotic behavior of the lower capacitive dimension of closed $\varepsilon$-neighborhoods of zero-dimensional, in the sense of $\dim_B$, closed subsets of $X$ as $\varepsilon\to 0$. For a wide class of metric compacta, the equality $z\underline{\dim}_BX=\underline{\dim}_BX$ holds.
Keywords: space of probability measures, theorem on intermediate values of the dimension of quantization.
Mots-clés : capacitive dimension, dimension of quantization 
@article{MZM_2024_115_3_a2,
     author = {A. V. Ivanov},
     title = {On the {Intermediate} {Values}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {348--354},
     year = {2024},
     volume = {115},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a2/}
}
TY  - JOUR
AU  - A. V. Ivanov
TI  - On the Intermediate Values
JO  - Matematičeskie zametki
PY  - 2024
SP  - 348
EP  - 354
VL  - 115
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a2/
LA  - ru
ID  - MZM_2024_115_3_a2
ER  - 
%0 Journal Article
%A A. V. Ivanov
%T On the Intermediate Values
%J Matematičeskie zametki
%D 2024
%P 348-354
%V 115
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a2/
%G ru
%F MZM_2024_115_3_a2
A. V. Ivanov. On the Intermediate Values. Matematičeskie zametki, Tome 115 (2024) no. 3, pp. 348-354. http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a2/

[1] S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Math., 1730, Springer-Verlag, Berlin, 2000 | MR

[2] V. V. Fedorchuk, “Bikompakty bez promezhutochnykh razmernostei”, Dokl. AN SSSR, 213:4 (1973), 795–797 | MR | Zbl

[3] A. V. Ivanov, “O mnozhestve znachenii razmernosti kvantovaniya veroyatnostnykh mer na metricheskom kompakte”, Sib. matem. zhurn., 63:5 (2022), 1074–1080 | DOI

[4] Ya. B. Pesin, Teoriya razmernosti i dinamicheskie sistemy: sovremennyi vzglyad i prilozheniya, In-t kompyuternykh issledovanii, M.–Izhevsk, 2013

[5] A. V. Ivanov, “On quantization dimensions of idempotent probability measures”, Topology Appl., 306:1 (2022), 107931 | MR

[6] V. V. Fedorchuk, V. V. Filippov, Obschaya topologiya. Osnovnye konstruktsii, Izd-vo MGU, M., 1988

[7] A. V. Ivanov, “O funktore veroyatnostnykh mer i razmernostyakh kvantovaniya”, Vestnik Tomsk. gos. un-ta. Mat. mekh., 63 (2020), 15–26 | MR