Positive Lyapunov exponent for a class of quasi-periodic cocycles

Jinhao Liang

doi:10.3934/dcds.2020080

Article Contents

2020, Volume 40, Issue 3: 1361-1387. Doi: 10.3934/dcds.2020080

This issue Previous Article Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps Next Article Existence of periodically invariant tori on resonant surfaces for twist mappings

Positive Lyapunov exponent for a class of quasi-periodic cocycles

Jinhao Liang

Department of Mathematics, Southeast University, Nanjing 211189, China

Received: December 31, 2018

Revised: September 30, 2019

Published: December 2019

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Abstract

Young [17] proved the positivity of Lyapunov exponent in a large set of the energies for some quasi-periodic cocycles. Her result is also proved to be applicable for some quasi-periodic Schrödinger cocycles by Zhang [18]. However, her result cannot be applied to the Schrödinger cocycles with the potential $ v = \cos(4\pi x)+w( x) $, where $ w\in C^2(\mathbb R/\mathbb Z,\mathbb R) $ is a small perturbation. In this paper, we will improve her result such that it can be applied to more cocycles.

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Mathematics Subject Classification: Primary: 37A30.

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Figure 1. graph of the function in $ \mathcal F $

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Figure 2. Graphs of the angle functions

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Figure 3. Bifurcation of Type Ⅲ functions with $ f'_1(c_1)f'_2(c_2)<0 $

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References

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