On the spectra of certain matrices and the iteration of quadratic maps

We study a sequence of one-parameter matrices, \(A_{k}\left( c\right) \), with \(k\in \mathbb {N}\) and parameter \(c\in \mathbb {C}\), which is obtained recursively. For the given recursion, the characteristic polynomial of \(A_{k}\left( c\right) \) is given by \(-f_{c}^{k}\left( x\right) \), where \(f_{c}\left( x\right) =c-x^{2}\) and \(f_{c}^{k}\left( x\right) =f_{c}\circ \cdots \circ f_{c}\left( x\right) \). Therefore, the spectrum of each matrix \(A_{k}\left( c\right) \) corresponds to the pre-images of \(0\) under iteration of \(f_{c}\). We obtain the block structure and a recursion for the eigenvectors of \(A_k(c),\ k=0,1,\ldots \). The structure of the eigenspaces of \(A_k(c)\) change dramatically with the parameter \(c\), in particular their dimension as real linear spaces. For \(c\in \mathbb {R}\), we extensively use symbolic dynamics for unimodal maps of an interval to study this problem. We show that the logarithm of the growth rate of the real eigenspaces dimensions is equal to the topological entropy of \(f_c\). Moreover, for \(c\in \mathbb {C}\), the relation between the sequence \(A_{k}\left( c\right) \) and the iteration of \(f_{c}\) gives us an interesting interpretation of the spectrum of \(A_{k}\left( c\right) \) as the Julia set of \(f_{c}\).