LOCALIZED BIRKHOFF AVERAGE IN BETA DYNAMICAL SYSTEMS

In this note, we investigate the localized multifractal spectrum of Birkhoff average in the beta-dynamical system ([0, 1]; T-beta) for general beta > 1, namely the dimension of the following level sets {x is an element of [0, 1]: lim(n ->infinity) 1/n Sigma(n-1)(j=0)psi(T(j)x) = f(x)}. where f and psi are two continuous functions defined on the unit interval [0, 1]. Instead of a constant function in the classical multifractal cases, the function f here varies with x. The method adopted in the proof indicates that the multifractal analysis of Birkhoff average in a general beta-dynamical system can be achieved by approximating the system by its subsystems.