### Abstract

In the current paper we address the problem of classification of cocycles over an irrational rotation. We use the renormalization group approach. A cocycle means a C^{r}-mapping u:T →SU (2, ℂ) (r≧2). We fix an irrational rotation τ:T →T. Two cocycles u, v:T →SU(2, ℂ) are considered equivalent (or cohomologous) if there is a continuous map h:T→SU (2, ℂ) such that u·h{o script} τ=h·v. By definition, our problem is to classify cocycles up to this equivalence relation. By introducing a suitable renormalization map we are able to define a notion of a fiber rotation number for a class of cocycles which are in the basin of the attractor of the renormalization map. The attractor itself is built of algebraic Anosov maps on T^{2}. We present a number of results and conjecuters resulting from this approach. We show how this approach sheds some light upon the problem of classifying linear ODE with almost-periodic, skew-hermitian coefficient matrix.

Original language | English (US) |
---|---|

Pages (from-to) | 173-206 |

Number of pages | 34 |

Journal | Inventiones Mathematicae |

Volume | 110 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1992 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Renormalization of cocycles and linear ODE with almost-periodic coefficients.** / Rychlik, Marek R.

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 110, no. 1, pp. 173-206. https://doi.org/10.1007/BF01231330

}

TY - JOUR

T1 - Renormalization of cocycles and linear ODE with almost-periodic coefficients

AU - Rychlik, Marek R

PY - 1992/12

Y1 - 1992/12

N2 - In the current paper we address the problem of classification of cocycles over an irrational rotation. We use the renormalization group approach. A cocycle means a Cr-mapping u:T →SU (2, ℂ) (r≧2). We fix an irrational rotation τ:T →T. Two cocycles u, v:T →SU(2, ℂ) are considered equivalent (or cohomologous) if there is a continuous map h:T→SU (2, ℂ) such that u·h{o script} τ=h·v. By definition, our problem is to classify cocycles up to this equivalence relation. By introducing a suitable renormalization map we are able to define a notion of a fiber rotation number for a class of cocycles which are in the basin of the attractor of the renormalization map. The attractor itself is built of algebraic Anosov maps on T2. We present a number of results and conjecuters resulting from this approach. We show how this approach sheds some light upon the problem of classifying linear ODE with almost-periodic, skew-hermitian coefficient matrix.

AB - In the current paper we address the problem of classification of cocycles over an irrational rotation. We use the renormalization group approach. A cocycle means a Cr-mapping u:T →SU (2, ℂ) (r≧2). We fix an irrational rotation τ:T →T. Two cocycles u, v:T →SU(2, ℂ) are considered equivalent (or cohomologous) if there is a continuous map h:T→SU (2, ℂ) such that u·h{o script} τ=h·v. By definition, our problem is to classify cocycles up to this equivalence relation. By introducing a suitable renormalization map we are able to define a notion of a fiber rotation number for a class of cocycles which are in the basin of the attractor of the renormalization map. The attractor itself is built of algebraic Anosov maps on T2. We present a number of results and conjecuters resulting from this approach. We show how this approach sheds some light upon the problem of classifying linear ODE with almost-periodic, skew-hermitian coefficient matrix.

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UR - http://www.scopus.com/inward/citedby.url?scp=0001512199&partnerID=8YFLogxK

U2 - 10.1007/BF01231330

DO - 10.1007/BF01231330

M3 - Article

AN - SCOPUS:0001512199

VL - 110

SP - 173

EP - 206

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -