Information Parameters and Large Deviations Spectrum of Discontinuous Measures

Let ν be a finite Borel measure on [0, 1]. Consider the L-spectrum of ν: τν(q) = lim infn→∞−n logb P Q∈Gn ν(Q) q (q ≥ 0), where Gn is the set of b-adic cubes of generation n (b integer ≥ 2). Let qτ = inf{q : τν(q) = 0} and Hτ = τ ′ ν(q τ ). When ν is a mono-dimensional continuous measure of information dimension D, (qτ , Hτ ) = (1, D). When ν is purely discontinuous, its information dimension is D = 0, but the pair (qτ , Hτ ) may be non-trivial and contains relevant information on the distribution of ν. Intrinsic characterizations of (qτ , Hτ ) are found, as well as sharp estimates for the large deviations spectrum of ν on [0, Hτ ]. We exhibit the differences between the cases qτ = 1 and qτ ∈ (0, 1). We conclude that the large deviations spectrum’s properties observed for specific classes of measures are true in general.