Published November 2012
| public
Journal Article
Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper's Model
- Creators
- Jitomirskaya, S.
- Marx, C. A.
Abstract
We show how to extend (and with what limitations) Avila's global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for the extended Harper's model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila's global theory is also shown to imply continuous behavior of the LE on the space of analytic M_2 (C)-cocycles. This includes rational approximation of the frequency, which so far has not been available.
Additional Information
© 2012 Springer-Verlag. Received: 14 September 2011; Accepted: 27 October 2011; Published online: 31 March 2012. The work was supported by the NSF Grants DMS-0601081 and DMS-1101578, and the BSF grant 2006483. Communicated by B. Simon. We are grateful to Artur Avila for his remarks on an earlier version of this paper where, among other things, he essentially provided a simple proof of continuity of the Lyapunov exponent of singular cocycles for all frequencies once such continuity for the Diophantine case is established, using the ideas of [2]. It turned out the same idea could be used to provide a simple proof of joint continuity (presented here), significantly simplifying our original approach. Additionally, our proof of continuity for the case of an identically vanishing determinant follows his suggestions as well.We also thank Anton Gorodetski for useful discussions during the preparation of this manuscript.Additional details
- Eprint ID
- 36907
- Resolver ID
- CaltechAUTHORS:20130214-075409144
- DMS-0601081
- NSF
- DMS-1101578
- NSF
- 2006483
- Binational Science Foundation (BSF)
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2013-02-26Created from EPrint's datestamp field
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2023-06-01Created from EPrint's last_modified field