Multifractal analysis of the divergence points of Birkhoff averages for β-transformations

This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of beta-expansions. More precisely, let T-beta be the beta-transformation on [0, 1) for a general beta > 1 and psi : [0, 1] bar right arrow R be a continuous function. Denote by A(psi, x) all the accumulation points of {1/n Sigma(n-1)(j=0) psi (T (j) x) : n >= 1}. The Hausdorff dimensions of the sets {x : A(psi, x) superset of [a, b]}, {x : A(psi, x) = [a, b]}, {x : A(psi, x) subset of [a, b]} i.e., the points for which the Birkhoff averages of psi do not exist but behave in a certain prescribed way, are determined completely for any continuous function psi.