Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, {_s, s S} be the semigroup of translations in the function space X = V^S, A = {A_n} be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X_f,,Adefined by the following formula: It turns out that this dimension does not depend on the choice of a Følner pointwise averaging sequence A and is completely specified by the scale index of the metric in X. This general model includes the important cases where , d 1, and the sets A_n are infinitely increasing cubes; if then f(x) = (f₁(x),..., f_m(x)rpar;, = (₁,...,_m), and Thus the multifractal analysis of the ergodic averages of several continuous functions is a special case of our results; in particular, in Examples 4 and 5 we generalize the well-known theorems due to Eggleston [3] and Billingsley [1].