Iteration of polynomial pair under Thue-Morse dynamic

Abstract

We study the behavior of a polynomial sequence which is defined by iterating a polynomial pair under Thue-Morse dynamic. We show that in suitable sense, the sequence will behave like $\{2\cos 2^{n}x:n\ge 1\}$. Basing on this property we can show that the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian has a common positive lower bound for all coupling.