We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach.

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关于无理圆旋转的三角斜积

我们研究了圆的无理旋转上三角斜积图的一些渐近特性。使用重整化控制限制。这里考虑的映射与能量为零的自对偶 Hofstadter 哈密顿量有关。它们类似于几乎 Mathieu 映射，但因子交换。这允许我们在重整化下为每个二次无理数构建周期轨道，并证明 Hofstadter 模型产生的映射对被这些周期轨道吸引。我们相信类似的结果适用于几乎属于 Mathieu 的自对偶映射，但它们目前似乎遥不可及。