Geodesic
On the structure of invariant measures for set-valued maps
Sbornik. Mathematics, Tome 202 (2011) no. 9, pp. 1285-1302 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Properties of measures invariant with respect to set-valued maps are studied. It is shown that an absolutely continuous invariant measure for a set-valued map need not be unique, and the set of all invariant measures need not be a Choquet simplex. The problem concerning the existence of invariant measures with respect to set-valued maps parametrized by single-valued and set-valued maps of the circle having various smoothness classes is studied. Bibliography: 13 titles.
Keywords: set-valued maps, invariant measure
Mots-clés : Choquet simplex. 
@article{SM_2011_202_9_a1,
     author = {A. N. Gorbachev and A. M. Stepin},
     title = {On the structure of invariant measures for set-valued maps},
     journal = {Sbornik. Mathematics},
     pages = {1285--1302},
     year = {2011},
     volume = {202},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2011_202_9_a1/}
}
TY  - JOUR
AU  - A. N. Gorbachev
AU  - A. M. Stepin
TI  - On the structure of invariant measures for set-valued maps
JO  - Sbornik. Mathematics
PY  - 2011
SP  - 1285
EP  - 1302
VL  - 202
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2011_202_9_a1/
LA  - en
ID  - SM_2011_202_9_a1
ER  - 
%0 Journal Article
%A A. N. Gorbachev
%A A. M. Stepin
%T On the structure of invariant measures for set-valued maps
%J Sbornik. Mathematics
%D 2011
%P 1285-1302
%V 202
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2011_202_9_a1/
%G en
%F SM_2011_202_9_a1
A. N. Gorbachev; A. M. Stepin. On the structure of invariant measures for set-valued maps. Sbornik. Mathematics, Tome 202 (2011) no. 9, pp. 1285-1302. http://geodesic.mathdoc.fr/item/SM_2011_202_9_a1/

[1] A. M. Vershik, “Polymorphisms, Markov processes, and quasi-similarity”, J. Soviet Math., 23:3 (1983), 2243–2266 | DOI | MR | Zbl | Zbl

[2] A. Lasota, J. A. Yorke, “On the existence of invariant measures for piecewise monotonic transformations”, Trans. Amer. Math. Soc., 186 (1973), 481–488 | DOI | MR | Zbl

[3] A. Réniy, “Representations for real numbers and their ergodic properties”, Acta Math. Acad. Sci. Hungar, 8:3–4 (1957), 477–493 | DOI | MR | Zbl

[4] V. A. Rokhlin, “Lectures on the entropy theory of measure-preserving transformations”, Russian Math. Surveys, 22:5 (1967), 1–52 | DOI | MR | Zbl

[5] N. N. Bogolubov, Selected works, v. I, Classics Soviet Math., 2, Gordon and Breach, New York, 1990 | MR | Zbl | Zbl

[6] M. Beboutoff, “Markoff chains with a compact state space”, Matem. sb., 10(52):3 (1942), 213–238 | MR | Zbl

[7] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand, London, 1969 | MR | Zbl | Zbl

[8] R. I. Grigorchuk, A. M. Stepin, “On amenability of semigroups with cancellation”, Moscow Univ. Math. Bull., 53:3 (1998), 7–11 | MR | Zbl

[9] I. Namioka, “Folner's conditions for amenable semi-groups”, Math. Scand., 15 (1964), 18–28 | MR | Zbl

[10] Sh. I. Akhalaya, A. M. Stepin, “On absolutely continuous invariant measures of noncontracting transformations of a circle”, Proc. Steklov Inst. Math., 244 (2004), 18–28 | MR | Zbl

[11] D. G. Kendall, “Simplexes and vector lattices”, J. London Math. Soc., 37:3 (1962), 365–371 | DOI | MR | Zbl

[12] R. R. Phelps, Lectures on Choquet's theorem, Van Nostrand, Princeton, NJ–Toronto, ON–London, 1966 | MR | Zbl | Zbl

[13] N. K. Bary, A treatise on trigonometric series, Macmillan, New York, 1964 | MR | MR | Zbl