The paper considers the discrete Schrödinger operator associated to a Sturmian quasiperiodic potential \[ (H\psi)(n)=\psi(n-1)-\psi(n+1)+v(n,\theta)\psi(n). \] The quasiperiodicity of \(v(n)\) is generated by an irrational number \(\theta\) according to the formula \[ v(n,\theta)=V([(n+1)\theta]-[n\theta]), \] where \(V\in\mathbb R^+\) is the coupling constant and \([x]\) denotes the largest integer smaller than \(x\). The attention is focused on the potential associated to the irrational number \(\omega\), the so-called silver ratio \[ \omega=\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots}}}=[0,2,2,\dots,2,\dots]. \] It is shown that under this restriction one can study the spectrum of \(H\) by means of an auxiliary dynamical system, described by the so-called trace map \[ T(x,y,z):=(x(y^2-1)-zy,xy-z,y). \] The spectrum of \(H\) can be determined by studing the non-wandering set of \(T\) on a certain invariant surface \(S_V\). It is shown that the non-wandering set of \(T\) is hyperbolic provided the coupling \(V\) is sufficiently large. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and global box counting dimension of the spectum of \(H\) coincide and are smooth functions of the coupling constant.