Geodesic
Invariant densities for random $\beta$-expansions
Journal of the European Mathematical Society, Tome 9 (2007) no. 1, pp. 157-176.

Voir la notice de l'article provenant de la source EMS Press

Let β>1 be a non-integer. We consider expansions of the form ∑i=1∞​βidi​​, where the digits (di​)i≥1​ are generated by means of a Borel map Kβ​ defined on {0,1}N×[0,⌊β⌋/(β−1)]. We show existence and uniqueness of an absolutely continuous Kβ​-invariant probability measure w.r.t. mp​⊗λ, where mp​ is the Bernoulli measure on {0,1}N with parameter p (01) and λ is the normalized Lebesgue measure on [0,⌊β⌋/(β−1)]. Furthermore, this measure is of the form mp​⊗μβ,p​, where μβ,p​ is equivalent with λ. We establish the fact that the measure of maximal entropy and mp​⊗λ are mutually singular. In case 1 has a finite greedy expansion with positive coefficients, the measure mp​⊗μβ,p​ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].
DOI : 10.4171/jems/76
Classification : 28-XX, 11-XX, 37-XX, 00-XX
Keywords: greedy expansions, lazy expansions, absolutely continuous invariant measures, measures of maximal entropy, Markov chains, universal expansions 
@article{JEMS_2007_9_1_a6,
     author = {Karma Dajani and Martijn de Vries},
     title = {Invariant densities for random $\beta$-expansions},
     journal = {Journal of the European Mathematical Society},
     pages = {157--176},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {2007},
     doi = {10.4171/jems/76},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/76/}
}
TY  - JOUR
AU  - Karma Dajani
AU  - Martijn de Vries
TI  - Invariant densities for random $\beta$-expansions
JO  - Journal of the European Mathematical Society
PY  - 2007
SP  - 157
EP  - 176
VL  - 9
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4171/jems/76/
DO  - 10.4171/jems/76
ID  - JEMS_2007_9_1_a6
ER  - 
%0 Journal Article
%A Karma Dajani
%A Martijn de Vries
%T Invariant densities for random $\beta$-expansions
%J Journal of the European Mathematical Society
%D 2007
%P 157-176
%V 9
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4171/jems/76/
%R 10.4171/jems/76
%F JEMS_2007_9_1_a6
Karma Dajani; Martijn de Vries. Invariant densities for random $\beta$-expansions. Journal of the European Mathematical Society, Tome 9 (2007) no. 1, pp. 157-176. doi : 10.4171/jems/76. http://geodesic.mathdoc.fr/articles/10.4171/jems/76/

Cité par Sources :