Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential

By Jean Bourgain, S. Jitomirskaya. 2001

217 citations recorded by the source corpus.

We study regularity properties of the Lyapunov exponent L of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove joint continuity of L, in frequency and energy, at every irrational frequency.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7ea697e38d381234ff9c9011406bd3ebe9da0b83

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In Applied Mathematics

1. David Damanik
2. Jiangong You
3. Qi Zhou
4. Barry Simon
5. Artur Avila
6. Marcelo Viana
7. Carlos Bocker-Neto
8. Jairo Bochi

In Mathematical Physics

1. Artur Avila
2. Marcelo Viana
3. David Damanik
4. Jiangong You
5. Qi Zhou
6. Carlos Bocker-Neto
7. Barry Simon
8. Jairo Bochi

Citations in the corpus

• Continuity of Lyapunov exponents for random two-dimensional matrices (2016, background): “Recently, Bourgain and Jitomirskaya [8, 9] proved continuous dependence of the exponents on the energy E , for one-dimensional quasiperiodic Schrödinger cocycles: Vn = V ( f n(θ)), where V : S1→ R is real-analytic and f is an irrational circle rotation.”
• Sharp phase transitions for the almost Mathieu operator (2015, methodology): “By Gordon’s lemma [21] and the exact formula of Lyapunov exponent [13], one can prove that Hλ,α,θ has purely singular continuous spectrum for any θ ∈ R if 1 < λ < e β 2 , see also Remark 3.”
• The Lyapunov exponents of generic volume-preserving and symplectic maps (2005, background): “See Ruelle [29] and also Bourgain, Jitomirskaya [9], [10] for a discussion and further references.”
• Almost reducibility and absolute continuity I (2010, background): “the problem of describing the stable regimes. As it turns out, the understanding of supercriticality is quite developed, through the works of Bourgain, Goldstein, Jitomirskaya and Schlag [BG], [GS1], [BJ1], [GS2], [GS3]. Subcriticality on the other hand, is a relatively new concept, which Date: Sept”
• EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY (2007, background): “E ∈ spec(J), and Bourgain–Jitomirskaya [19] proved continuity of γ.”
• Schrödinger operators with dynamically defined potentials (2014, result): “This estimate in turn is sharp, as shown by Bourgain and Jitomirskaya in [41]:”