On trigonometric skew-products over irrational circle-rotations

Hans Koch

doi:10.3934/dcds.2021084

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2021, Volume 41, Issue 11: 5455-5471. Doi: 10.3934/dcds.2021084

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On trigonometric skew-products over irrational circle-rotations

Hans Koch

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA

Received: April 30, 2020

Revised: February 28, 2021

Early access: May 2021

Published: November 2021

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Abstract

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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Mathematics Subject Classification: Primary: 37E20; Secondary: 37F25, 37N20.

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Figure 1. The orbit $ j\mapsto y^s_j $ for the inverse golden mean (left) and its absolute value (log scale, right)

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