Let \({\mathcal A}=\{ a, b\}\) and let \({\mathcal A}^{\star}\) denote the set of finite words over \({\mathcal A}\). The Fibonacci substitution \(S: {\mathcal A} \rightarrow {\mathcal A}^{\star}\) is defined by \(S: a\mapsto ab\), \(b\mapsto a\). The map \(S\) is extended to \({\mathcal A}^{\star}\) and \({\mathcal A}^{\mathbb N, \mathbb Z}\). There exists a unique substitution sequence \(u\in {\mathcal A}^{\mathbb N}\) with some properties using some sequence of Fibonacci numbers. Let \(\hat u\) denote an arbitrary extension of \(u\) to a two-sided sequence in \({\mathcal A}^{\mathbb Z}\). Define \(\Omega =\{ \omega \in {\mathcal A}^{\mathbb Z} : \omega =\lim_{i \rightarrow \infty} T^{n_i} (\cap u)\), \(n_i \uparrow \infty \}\), \(T: {\mathcal A}^{\mathbb Z} \rightarrow {\mathcal A}^{\mathbb Z}\), where \([T(v)]_n =v_{n+1}\). To each \(\omega \in \Omega\) is associated a Jacobi operator.
Main results of the paper. Theorem 2.1. There exists \(\Sigma_{(p,q)} \subset\mathbb R\) such that, for all \(\omega \in \Omega\), \(\sigma (H_{\omega} )= \Sigma_{(p,q)}\). If \((p,q) \neq (1,0)\), then \(\Sigma_{(p,q)}\) is a Cantor set of zero Lebesque measure; it is purely singular continuous. Theorem 2.3. For all \((p,q) \neq (1,0)\), the spectrum \(\Sigma_ {(p,q)}\) is a multifractal. Existence of box-counting dimension, when it exists whether it coincides with Hausdorff dimension, is of interest. In the paper, there is a partial answer in this direction. A theorem is proved that states that the point-wise dimension of \(dN\) exists \(dN\)-almost everywhere.