Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients. I: The essential support of the measure (Q390536)

Each probability measure \(\mu\) on the unit circle \(\partial{\mathbb D}\) corresponds to a unique sequence of normalized orthogonal polynomials \(\{\varphi_n\}_{n\geq0}\). This sequence coupled with the sequence of the ``reflected polynomials'' \(\{\varphi_n^*\}_{n\geq0}\) satisfies a particular vector recurrence relation depending on the spectral variable \(w\) and complex numbers \(\alpha_n\), \(|\alpha_n|<1\), called the Verblunsky coefficients.NEWLINENEWLINEThe authors study the 2-parameter family of the measures \(\mu_{\omega_{\alpha,\beta}}\) corresponding to the 2-parameter family of the Verblunsky coefficient sequences \(\{\alpha_n\}_{n\geq0}\), \(\alpha_n\in\{\alpha,\beta\}\), invariant under the Fibonacci substitution \(\alpha\mapsto\alpha\beta\), \(\beta\mapsto\alpha\). The main object of the study are the \(n\) large properties of the so-called trace map \(x_n(w)\). Essentially, the latter is the trace of the \(0\)- to \(k\)-transfer matrix corresponding to the vector recurrence relation and to the Fibonacci number \(k=f_n\), \(n\geq1\). The bounded sequences \(\{x_n(w)\}_{n\geq0}\) correspond to the points of the essential support of \(\mu_{\omega_{\alpha,\beta}}\). Using methods of dynamical systems, the authors prove that this support is a Cantor set of zero Lebesgue measure. They also study various fractal properties of the essential support of these measures including estimates for the local Hausdorff dimensions. Finally, they find the estimates for the growth rate of the transfer matrices and the orthogonal polynomials.