Multifractal Analysis Of The Divergence Points Of Birkhoff Averages In ?-dynamical Systems

Different from the systems with Markov properties, the research framework of this paper is based on the ?-expansions which is a non-Markov property for a general ?> 1. In order to have a better understanding of the non-Markov property and find ways to conquer the difficulties caused by it, this paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of ?-expansions. Having at last got the results of dimension of divergence sets given by Olsen, the ideas to prove the first item in the Theorem are similar to them, but for the second item, compares to the large deviation theory which they used, our solution is a simple Lebesgue covering lemma. More precisely, let ([0,1], T? be the ?-dynamical system for a general ?> 1 and ?:[0,1]?R be a continuous function. Denote by A(?,x) all the accumulation points of The Hausdorff dimensions of the sets(?)i.e., the points for which the Birkhoff averages of ? do not exist but behave in a certain prescribed way, are determined completely for any continuous function ?.For further studies, besides the research achievements of this paper, we could also attempt to do some explorations for the sets of Birkhoff averages diverge in the system of infinite iterated functions or other research backgrounds, for the sake of a better understanding.