scholarly journals Lyapunov exponents of random walks in small random potential: the upper bound

2015 ◽  
Vol 20 (0) ◽  
Author(s):  
Thomas Mountford ◽  
Jean-Christophe Mourrat
Keyword(s):  
Random Walks ◽  
Lyapunov Exponents ◽  
Upper Bound ◽  
Random Potential

scholarly journals Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective

2014 ◽  
Vol 15 (3) ◽  
pp. 712-732
Author(s):  
Eric Bourgain-Chang
Keyword(s):  
Lyapunov Exponents ◽  
Spectral Theory ◽  
Upper Bound ◽  
Shift Operator ◽  
Numerical Study ◽  
Striking Difference ◽  
Spectral Features ◽  
Random Models

AbstractIn this paper we perform a numerical study of the spectra, eigenstates, and Lyapunov exponents of the skew-shift counterpart to Harper’s equation. This study is motivated by various conjectures on the spectral theory of these ‘pseudo-random’ models, which are reviewed in detail in the initial sections of the paper. The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model. In particular our numerics establish a small upper bound on the gaps in the spectrum (conjectured to be absent).

scholarly journals C1-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents

2009 ◽  
Vol 9 (1) ◽  
pp. 49-93 ◽  
Author(s):  
Jairo Bochi
Keyword(s):  
Random Walks ◽  
Lyapunov Exponents ◽  
Probabilistic Method ◽  
International Congress ◽  
Partial Hyperbolicity ◽  
Symplectic Diffeomorphisms ◽  
Symplectic Diffeomorphism ◽  
Partially Hyperbolic

AbstractWe prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Mañé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

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