On Sl(2,ℝ) Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Mahesh Nerurkar

doi:10.1007/s10884-011-9215-4

On Sl(2,ℝ) Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Mahesh Nerurkar

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Consider the class of C^r-smooth SL(2,ℝ) valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition, where C > 0 and 0< κ< are some constants and P_n/q_n is some sequence of irreducible fractions.

Original language	English (US)
Pages (from-to)	451-473
Number of pages	23
Journal	Journal of Dynamics and Differential Equations
Volume	23
Issue number	3
DOIs	https://doi.org/10.1007/s10884-011-9215-4
State	Published - Sep 2011

All Science Journal Classification (ASJC) codes

Analysis

Keywords

Cocycles
Fast periodic approximation
Irrational rotations
Lyapunov exponents
Proximal extensions

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10.1007/s10884-011-9215-4

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abstract = "Consider the class of Cr-smooth SL(2,ℝ) valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition, where C > 0 and 0< κ< are some constants and Pn/qn is some sequence of irreducible fractions.",

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N2 - Consider the class of Cr-smooth SL(2,ℝ) valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition, where C > 0 and 0< κ< are some constants and Pn/qn is some sequence of irreducible fractions.

AB - Consider the class of Cr-smooth SL(2,ℝ) valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition, where C > 0 and 0< κ< are some constants and Pn/qn is some sequence of irreducible fractions.

KW - Cocycles

KW - Fast periodic approximation

KW - Irrational rotations

KW - Lyapunov exponents

KW - Proximal extensions

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JO - Journal of Dynamics and Differential Equations

JF - Journal of Dynamics and Differential Equations

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ER -

On Sl(2,ℝ) Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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