Previous |  Up |  Next

Article

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
Summary:
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.
References:
[1] Berend, D.: Multi-invariant sets on compact abelian groups. Trans. Am. Math. Soc. 286 (1984), 505-535. DOI 10.1090/S0002-9947-1984-0760973-X | MR 0760973 | Zbl 0523.22004
[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 1187044 | Zbl 0772.11041
[3] Chothi, V., Everest, G., Ward, T.: {$S$}-integer dynamical systems: periodic points. J. Reine Angew. Math. 489 (1997), 99-132. MR 1461206 | Zbl 0879.58037
[4] Drmota, M., Tichy, R. F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651 Springer, Berlin (1997). MR 1470456 | Zbl 0877.11043
[5] Gouvêa, F. Q.: $p$-adic Numbers: An Introduction. Universitext Springer, Berlin (1997). MR 1488696 | Zbl 0874.11002
[6] Halmos, P. R.: On automorphisms of compact groups. Bull. Am. Math. Soc. 49 (1943), 619-624. DOI 10.1090/S0002-9904-1943-07995-5 | MR 0008647 | Zbl 0061.04403
[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 0551496
[8] Host, B.: Some results of uniform distribution in the multidimensional torus. Ergodic Theory Dyn. Syst. 20 (2000), 439-452. MR 1756978 | Zbl 1047.37003
[9] Host, B.: Normal numbers, entropy, translations. Isr. J. Math. 91 French (1995), 419-428. DOI 10.1007/BF02761660 | MR 1348326 | Zbl 0839.11030
[10] Koblitz, N.: $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics 58 Springer, New York (1977). DOI 10.1007/978-1-4684-0047-2 | MR 0754003 | Zbl 0364.12015
[11] Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. John Wiley & Sons New York (1974). MR 0419394
[12] Mahler, K.: $p$-adic Numbers and Their Functions. Cambridge Tracts in Mathematics 76 Cambridge University Press, Cambridge (1981). MR 0644483 | Zbl 0444.12013
[13] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics Springer, Berlin (2004). MR 2078267 | Zbl 1159.11039
[14] Neukirch, J.: Algebraic Number Theory. Fundamental Principles of Mathematical Sciences 322 Springer, Berlin (1999). MR 1697859 | Zbl 0956.11021
[15] Ramakrishnan, D., Valenza, R. J.: Fourier Analysis on Number Fields. Graduate Texts in Mathematics 186 Springer, New York (1999). DOI 10.1007/978-1-4757-3085-2 | MR 1680912 | Zbl 0916.11058
[16] Robert, A. M.: A Course in $p$-adic Analysis. Graduate Texts in Mathematics 198 Springer, New York (2000). DOI 10.1007/978-1-4757-3254-2 | MR 1760253 | Zbl 0947.11035
[17] Schmidt, K.: Dynamical Systems of Algebraic Origin. Progress in Mathematics 128 Birkhäuser, Basel (1995). MR 1345152 | Zbl 0833.28001
[18] Urban, R.: Equidistribution in the {$d$}-dimensional {$a$}-adic solenoids. Unif. Distrib. Theory 6 (2011), 21-31. MR 2817758
[19] Weil, A.: Basic Number Theory. Die Grundlehren der Mathematischen Wissenschaften 144 Springer, New York (1974). MR 0427267 | Zbl 0326.12001
Partner of
EuDML logo