Overlap-free words and spectra of matrices

Abstract

Overlap-free words are words over the binary alphabet $A=\{a, b\}$ that do not contain factors of the form $xvxvx$, where $x \in A$ and $v \in A^*$. We analyze the asymptotic growth of the number $u_n$ of overlap-free words of length $n$ as $ n \to \infty$. We obtain explicit formulas for the minimal and maximal rates of growth of $u_n$ in terms of spectral characteristics (the lower spectral radius and the joint spectral radius) of certain sets of matrices of dimension $20 \times 20$. Using these descriptions we provide new estimates of the rates of growth that are within 0.4% and $0.03 %$ of their exact values. The best previously known bounds were within 11% and 3% respectively. We then prove that the value of $u_n$ actually has the same rate of growth for ``almost all'' natural numbers $n$. This ``average'' growth is distinct from the maximal and minimal rates and can also be expressed in terms of a spectral quantity (the Lyapunov exponent). We use this expression to estimate it. In order to obtain our estimates, we introduce new algorithms to compute spectral characteristics of sets of matrices. These algorithms can be used in other contexts and are of independent interest.