Multifractal analysis of ergodic averages: A generalization of eggleston's theorem

Let V be a finite set, S be an infinite countable commutative semigroup, {τ_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 α ∈ 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_α,f,A defined by the following formula: X_α,f,A = {x : x ∈ X, lim_n→∞ 1/|A_n| ∑_s∈A(n) f(τ_sx) = α}. 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 special cases where S = ℤ^d or ℤ₊^d, d ≥ 1, and the sets A_n are infinitely increasing cubes; if B = ℝ^m then f(x) = (f₁(x), ..., f_m(x)), α = (α₁, ..., α_m), and X_α,f,A = {x : x ∈ X, lim_n→∞ 1/|A_n| ∑_s∈A(n) f₁(τ_sx) = = α₁, ..., lim _n→∞ 1/|A_n| ∑_s∈A(n) f_m(τ_sx) = α_m}. 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].