Iteration of polynomial pair under Thue-Morse dynamic

TL;DR

This paper investigates the iterative behavior of polynomial pairs under Thue-Morse dynamics and demonstrates a connection to cosine sequences, leading to a lower bound on the spectrum's Hausdorff dimension for the Thue-Morse Hamiltonian.

Contribution

It introduces a novel analysis of polynomial iteration under Thue-Morse dynamics and links it to spectral properties of the associated Hamiltonian.

Findings

Sequence behaves like rac{2 ext{cos} 2^n x}{n ext{large}}",

Hausdorff dimension of spectrum has a positive lower bound for all couplings

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