On trigonometric skew-products over irrational circle-rotations

Koch, Hans

doi:10.3934/dcds.2021084

Cited by 5 publications

(13 citation statements)

References 23 publications

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“…The same applies to the iteration (B • , A • ) → B• , Ã• , except that the circle gets enlarged by a factor α −3 at each step. Furthermore, the zeros that lie within some fixed positive angle of the origin reproduce after each step [54]. This yields the limiting sequences of zeros described in Theorem 2.4.…”

Section: The Supercritical Fixed Pointmentioning

confidence: 76%

“…This conjecture has motivated the work presented in this paper as well as our earlier work in [51,52,54]. The integers ℓ that appear in (1.6) can be obtained by considering the map on the torus T 2 given by the matrix 1 1 1 0 .…”

Section: Introductionmentioning

confidence: 78%

“…By contrast, proving that the AM family and others are attracted to the unstable manifold of R ℓ at P * is a global analysis and much harder. A simplified version of this problem was considered in [54], in a situation where the Hofstadter Hamiltonian reduces to skew-product maps with factors that take values in the circle R ∪ {∞}. Here we prove convergence (1.6) with SL(2, R) factors, including the AM factors, but only in a restricted one-parameter setup.…”

Section: Introductionmentioning

confidence: 82%

“…By controlling the zero sets B n and A n , one finds uniform convergence on compact sets to a pair of limit functions b ⋄ and a ⋄ . For details we refer to [54], where RG fixed points have been constructed for skew-products with meromorphic factors.…”

Section: The Supercritical Fixed Pointmentioning

confidence: 99%

“…Proof of Theorem 2.4. We give only a sketch here, since the proof follows closely the steps used in [54].…”

mentioning

confidence: 99%

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Asymptotic scaling and universality for skew products with factors in SL(2,R)

Koch

2021

Preprint

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We consider skew-product maps over circle rotations x → x+α (mod 1) with factors that take values in SL(2, R). This includes maps of almost Mathieu type. In numerical experiments, with α the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

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Section: The Supercritical Fixed Pointmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 82%

Section: The Supercritical Fixed Pointmentioning

confidence: 99%

“…Proof of Theorem 2.4. We give only a sketch here, since the proof follows closely the steps used in [54].…”

mentioning

confidence: 99%

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Asymptotic scaling and universality for skew products with factors in SL(2,R)

Koch

2021

Preprint

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Asymptotic scaling and universality for skew products with factors in SL(2,)

Koch

2022

Ergod. Th. Dynam. Sys.

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We consider skew-product maps over circle rotations $x\mapsto x+\alpha \;(\mod 1)$ with factors that take values in ${\textrm {SL}}(2,{\mathbb {R}})$ . In numerical experiments, with $\alpha $ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

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Renormalization and universality of the Hofstadter spectrum

Koch

Kocic

2020

Nonlinearity

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We consider a renormalization transformation R for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of R determine universal constants analogous to the critical exponents in the theory of phase transitions. Restricting to skew-product maps over circlerotations by the golden mean, we nd several periodic orbits for R, and we conjecture that there are in nitely many. Interestingly, all scaling factors that have been determined to high accuracy appear to be algebraically related to the circle-rotation number. We present evidence that these values describe (among other things) local scaling properties of the Hofstadter spectrum.

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On trigonometric skew-products over irrational circle-rotations

Cited by 5 publications

References 23 publications

Asymptotic scaling and universality for skew products with factors in SL(2,R)

Asymptotic scaling and universality for skew products with factors in SL(2,R)

Asymptotic scaling and universality for skew products with factors in SL(2,)

Renormalization and universality of the Hofstadter spectrum

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