$L^q$-spectrum of the Bernoulli convolution associated with the golden ratio
Based on the higher order self-similarity of the Bernoulli convolution measure for (p 5?1)=2 proposed by Strichartz et al, we derive a formula for the L q-spectrum, q > 0 of the measure. This formula is the rst one obtained in the case where the open set condition does not hold. x1. Introduction. Let be a positive bounded regular Borel measure on R d with compact support. For h > 0 and q > 0; we deene the L q-(moment) spectrum of by (q) = lim h!0 + ln P i (Q i (h)) q ln h (1.1) where fQ i (h)g i is a family of h-mesh cubes. We also deene the (lower) L q-dimension of by dim q () = (q)=(q ? 1); q > 1: These notions were rst used by Renyi R e] to extend the entropy dimension (corresponding to q = 1). For some variants of the deenition one can refer to St], LN1]. We