EuDML | Browse

Page 1

Displaying 1 – 12 of 12

Harmonic measures versus quasiconformal measures for hyperbolic groups

Sébastien Blachère, Peter Haïssinsky, Pierre Mathieu (2011)

Annales scientifiques de l'École Normale Supérieure

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.

Harmonic properties of some arithmetical sequences.

Jean Coquet (1982)

Manuscripta mathematica

Hausdorff dimension and a generalized form of simultaneous Diophantine approximation

B. P. Rynne, H. Dickinson (2000)

Acta Arithmetica

Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation.

M.M. Dodson (1992)

Journal für die reine und angewandte Mathematik

Hausdorff dimension of real numbers with bounded digit averages

Eda Cesaratto, Brigitte Vallée (2006)

Acta Arithmetica

Hausdorff dimension of the maximal run-length in dyadic expansion

Ruibiao Zou (2011)

Czechoslovak Mathematical Journal

For any $x \in [0, 1)$ , let $x = [ϵ_{1}, ϵ_{2}, \dots,]$ be its dyadic expansion. Call $r_{n} (x) : = max {j \geq 1 : ϵ_{i + 1} = \dots = ϵ_{i + j} = 1$ , $0 \leq i \leq n - j}$ the $n$ -th maximal run-length function of $x$ . P. Erdös and A. Rényi showed that ${lim}_{n \to \infty} r_{n} (x) / {log}_{2} n = 1$ almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than ${log}_{2} n$ , is quantified by their Hausdorff dimension.