It is known that the Lyapunov exponent of analytic 1-frequency quasiperiodic cocycles is continuous in cocycle and, when the frequency is irrational, jointly in cocycle and frequency. In this paper, we extend a result of Bourgain to show the same continuity result for multifrequency quasiperiodic $M(2,\mathbb{C})$ cocycles. Our corollaries include applications to multifrequency Jacobi cocycles with periodic background potentials.
Differentiability of Effective Fronts in the Continuous Setting in Two Dimensions
We study the effective front associated with first-order front propagations in two dimensions ($n=2$) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous ${\mathbb{Z}}^{2}$-periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for smooth metrics [ 4, 6]. To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the literature of partial differential equations (PDE). Combining with the sufficiency result in [ 15], our result leads to a realization type conclusion: for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior.
more » « less- Award ID(s):
- 1843320
- PAR ID:
- 10539361
- Publisher / Repository:
- Oxford Academic Group
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 7
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 5548 to 5585
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnad212
