  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 $\{2\cos 2^nx: n\ge 1\}$. 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.
