Geodesic
Equidistribution in the dual group of the $S$-adic integers
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 911-931
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu $ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n\bold {x}$ for $\mu $-almost all $\bold {x}\in X^d$ and the behavior of $\mu $ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus $\mathbb T^d$ due to B. Host.
Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu $ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n\bold {x}$ for $\mu $-almost all $\bold {x}\in X^d$ and the behavior of $\mu $ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus $\mathbb T^d$ due to B. Host.
DOI : 10.1007/s10587-014-0143-4
Classification : 11J71, 11K06, 54H20
Keywords: uniform distribution modulo $1$; equidistribution in probability; algebraic number fields; $S$-adele ring; $S$-integer dynamical system; algebraic dynamics; topological dynamics; $a$-adic solenoid 
@article{10_1007_s10587_014_0143_4,
     author = {Urban, Roman},
     title = {Equidistribution in the dual group of the $S$-adic integers},
     journal = {Czechoslovak Mathematical Journal},
     pages = {911--931},
     year = {2014},
     volume = {64},
     number = {4},
     doi = {10.1007/s10587-014-0143-4},
     mrnumber = {3304788},
     zbl = {06433704},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0143-4/}
}
TY  - JOUR
AU  - Urban, Roman
TI  - Equidistribution in the dual group of the $S$-adic integers
JO  - Czechoslovak Mathematical Journal
PY  - 2014
SP  - 911
EP  - 931
VL  - 64
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0143-4/
DO  - 10.1007/s10587-014-0143-4
LA  - en
ID  - 10_1007_s10587_014_0143_4
ER  - 
%0 Journal Article
%A Urban, Roman
%T Equidistribution in the dual group of the $S$-adic integers
%J Czechoslovak Mathematical Journal
%D 2014
%P 911-931
%V 64
%N 4
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0143-4/
%R 10.1007/s10587-014-0143-4
%G en
%F 10_1007_s10587_014_0143_4
Urban, Roman. Equidistribution in the dual group of the $S$-adic integers. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 911-931. doi: 10.1007/s10587-014-0143-4

[1] Berend, D.: Multi-invariant sets on compact abelian groups. Trans. Am. Math. Soc. 286 (1984), 505-535. | DOI | MR | Zbl

[2] Bertin, M.-J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., Schreiber, J.-P.: Pisot and Salem Numbers. With a preface by David W. Boyd Birkhäuser Basel (1992). | MR | Zbl

[3] Chothi, V., Everest, G., Ward, T.: {$S$}-integer dynamical systems: periodic points. J. Reine Angew. Math. 489 (1997), 99-132. | MR | Zbl

[4] Drmota, M., Tichy, R. F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651 Springer, Berlin (1997). | DOI | MR | Zbl

[5] Gouvêa, F. Q.: $p$-adic Numbers: An Introduction. Universitext Springer, Berlin (1997). | MR | Zbl

[6] Halmos, P. R.: On automorphisms of compact groups. Bull. Am. Math. Soc. 49 (1943), 619-624. | DOI | MR | Zbl

[7] Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis. Vol. I: Structure of topological groups, integration theory, group representations. Fundamental Principles of Mathematical Sciences 115 Springer, Berlin (1979). | MR

[8] Host, B.: Some results of uniform distribution in the multidimensional torus. Ergodic Theory Dyn. Syst. 20 (2000), 439-452. | MR | Zbl

[9] Host, B.: Normal numbers, entropy, translations. Isr. J. Math. 91 French (1995), 419-428. | DOI | MR | Zbl

[10] Koblitz, N.: $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics 58 Springer, New York (1977). | DOI | MR | Zbl

[11] Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. John Wiley & Sons New York (1974). | MR

[12] Mahler, K.: $p$-adic Numbers and Their Functions. Cambridge Tracts in Mathematics 76 Cambridge University Press, Cambridge (1981). | MR | Zbl

[13] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics Springer, Berlin (2004). | MR | Zbl

[14] Neukirch, J.: Algebraic Number Theory. Fundamental Principles of Mathematical Sciences 322 Springer, Berlin (1999). | MR | Zbl

[15] Ramakrishnan, D., Valenza, R. J.: Fourier Analysis on Number Fields. Graduate Texts in Mathematics 186 Springer, New York (1999). | DOI | MR | Zbl

[16] Robert, A. M.: A Course in $p$-adic Analysis. Graduate Texts in Mathematics 198 Springer, New York (2000). | DOI | MR | Zbl

[17] Schmidt, K.: Dynamical Systems of Algebraic Origin. Progress in Mathematics 128 Birkhäuser, Basel (1995). | MR | Zbl

[18] Urban, R.: Equidistribution in the {$d$}-dimensional {$a$}-adic solenoids. Unif. Distrib. Theory 6 (2011), 21-31. | MR

[19] Weil, A.: Basic Number Theory. Die Grundlehren der Mathematischen Wissenschaften 144 Springer, New York (1974). | MR | Zbl

Cité par Sources :