# Some generic properties of concentration dimension of measure

Bollettino dell'Unione Matematica Italiana (2003)

- Volume: 6-B, Issue: 1, page 211-219
- ISSN: 0392-4041

## Access Full Article

top## Abstract

top## How to cite

topMyjak, Józef, and Szarek, Tomasz. "Some generic properties of concentration dimension of measure." Bollettino dell'Unione Matematica Italiana 6-B.1 (2003): 211-219. <http://eudml.org/doc/195320>.

@article{Myjak2003,

abstract = {Let $K$ be a compact quasi self-similar set in a complete metric space $X$ and let $\mathfrak\{M\}_\{1\}(K)$ denote the space of all probability measures on $K$, endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in $\mathfrak\{M\}_\{1\}(K)$ the lower concentration dimension is equal to $0$, while the upper concentration dimension is equal to the Hausdorff dimension of $K$.},

author = {Myjak, Józef, Szarek, Tomasz},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {2},

number = {1},

pages = {211-219},

publisher = {Unione Matematica Italiana},

title = {Some generic properties of concentration dimension of measure},

url = {http://eudml.org/doc/195320},

volume = {6-B},

year = {2003},

}

TY - JOUR

AU - Myjak, Józef

AU - Szarek, Tomasz

TI - Some generic properties of concentration dimension of measure

JO - Bollettino dell'Unione Matematica Italiana

DA - 2003/2//

PB - Unione Matematica Italiana

VL - 6-B

IS - 1

SP - 211

EP - 219

AB - Let $K$ be a compact quasi self-similar set in a complete metric space $X$ and let $\mathfrak{M}_{1}(K)$ denote the space of all probability measures on $K$, endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in $\mathfrak{M}_{1}(K)$ the lower concentration dimension is equal to $0$, while the upper concentration dimension is equal to the Hausdorff dimension of $K$.

LA - eng

UR - http://eudml.org/doc/195320

ER -

## References

top- FALCONER, K. J., Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc., 106 (1989), 543-554. Zbl0683.58034MR969315
- GENYUK, J., A typical measure typically has no local dimension, Real Anal. Exchange, 23 (1997/1998), 525-537. Zbl0943.28008MR1639964
- GRUBER, P. M., Dimension and structure of typical compact sets, continua and curves, Mh. Math., 108 (1989), 149-164. Zbl0666.28005MR1026615
- HENGARTNER, W.- THEODORESCU, R., Concentration Functions, Academic Press, New York-London (1973). Zbl0323.60015MR331448
- HUTCHINSON, J. E., Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. Zbl0598.28011MR625600
- LASOTA, A.- MYJAK, J., On a dimension of measures, Bull. Pol. Ac. Math.2002. Zbl1020.28004
- MCLAUGHLIN, J., A note on Hausdorff measures of quasi self-similar sets, Proc. Amer. Math. Soc., 100 (1987), 183-186. Zbl0629.28006MR883425
- MYJAK, J.- RUDNICKI, R., Box and packing dimension of typical compact sets, Monatsh. Math., 131 (2000), 223-226. Zbl0967.28003MR1801749
- MYJAK, J.- RUDNICKI, R., On the typical structure of compact sets, Arch. Math., 76 (2001), 119-126. Zbl0981.46018MR1811289
- MYJAK, J.- RUDNICKI, R., Typical properties of correlation dimension (to appear). Zbl1048.37020MR2009754
- MYJAK, J.- SZAREK, T., Szpilrajn type theorem for concentration dimension of measure, Fund. Math., 172 (2002), 19-25. Zbl0994.37011MR1898400
- SULLIVAN, D., Seminar on conformal and hyperbolic geometry, Lecture Notes, Inst. Hautes Etudes Sci., Bures-sur-Yvette1982.
- MYJAK, J.- RUDNICKI, R., On the Box Dimension of Typical Measures, Monatsh. Math., 136 (2002), 143-150. Zbl1001.28002MR1914225
- MYJAK, J.- SZARCK, T., Generic properties of Markov operators, Rend. Circ. Matem. Palermo (to appear). Zbl1118.37011

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.