"Mixed spectral nature" of Thue--Morse Hamiltonian

We find three dense subsets $Σ_I,Σ_{II}$ and $Σ_{III}$ of the spectrum of the Thue-Morse Hamiltonian, such that each energy in $Σ_I$ is extended, each energy in $Σ_{II}$ is pseudo-localized and each energy in $Σ_{III}$ is one-sided pseudo-localized. We also obtain exact estimations on the norm of the transfer matrices and the norm of the formal solutions for these energies. Especially, for $E\in Σ_{II}\cupΣ_{III},$ the norms of the transfer matrices behave like $e^{c_1γ\sqrt{n}}\le\|T_{n}(E)\|\le e^{c_2γ\sqrt{n}}.$ The local dimensions of the spectral measure on these subsets are also studied. The local dimension is $0$ for energy in $Σ_{II}$ and is larger than $1$ for energy in $Σ_{I}\cupΣ_{III}.$ In summary, the Thue-Morse Hamiltonian exhibits "mixed spectral nature".

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