Small sets of divergence points are dimensionless

Let Sigma be a subshift modelled by a strongly connected graph, and let S:Sigma -->Sigma denote the shift. For n is an element of N, let L-n: Sigma --> P(Sigma) be the n'th empirical measure, i.e.

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where delta(x) denotes the Dirac measure at x and P(Sigma) denotes the family of probability measures on Sigma. We consider continuous deformations of L-n, i.e. pairs (X, Xi) where X is a metric vector space and Xi: P(Sigma) --> X is continuous with respect to the weak topology on P(Sigma). For a sequence (x(n))(n) in X, we write A(x(n)) for the set of accumulation points of (x(n))(n). Assume that T and U are subsets of X, and define

Delta(T, U) = {omega is an element of Sigma\Tsubset of or equal toA(XiL(n)omega) subset of or equal toU}.

In previous work we computed the Hausdorff dimension of the sets Delta(T,U). In this paper we prove that these sets are dimensionless, i.e. if t denotes the Hausdorff dimension of Delta(T,U) and h is a dimension function of the form h(r) = r(t)L(t) where L is a slowly varying function, then

H-h(Delta(T,U)) = 0 or H-h(Delta(T,U)) = infinity,

provided that t < dim &USigma;. In particular, H-t(&UDelta;(T,U)) = 0 or H-t(&UDelta;(T,U)) = &INFIN;. This implies that various sets of divergence points associated with different multifractal spectra (e.g. local dimensions, local entropies, local Lyapunov exponents, ergodic averages) are dimensionless.