Gibbs-like measure for spectrum of a class of quasi-crystals

Let αϵ(0,1) be an irrational, and [0;a 1,a 2,... ] the continued fraction expansion of α. Let H α,V be the one-dimensional Schrödinger operator with Sturmian potential of frequency α. Suppose the potential strength V >20 and the sequence (a i) i≥1 is bounded. We proceed by developing some new ideas on dimensional theory of Cookie-cutter sets. We prove that the spectral generating bands satisfy the principles of bounded variation and bounded covariation, and then we show that there exists a Gibbs-like measure on the spectrum σ(H α,V). As an application, we prove that dim H σ(H{α ,V})=s∗, where s ∗ and s ^∗ are the lower and upper pre-dimensions. Moreover, if (a n) n≥1 is ultimately periodic, then s ∗ =s ^∗.