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On transfer operators for continued fractions with restricted digits
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Jenkinson, Oliver, Gonzalez, Luis Felipe and Urba'nski, Mariusz. (2003) On transfer operators for continued fractions with restricted digits. Proceedings of the London Mathematical Society, Vol.86 (No.3). pp. 755-778. ISSN 0024-6115
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Official URL: http://dx.doi.org/10.1112/S0024611502013904
Abstract
For any non-empty subset I of the natural numbers, let {Lambda}I denote those numbers in the unit interval whose continued fraction digits all lie in I. Define the corresponding transfer operator Formula. for Formula, where Re (rß) = {theta}I is the abscissa of convergence of the series Formula. When acting on a certain Hilbert space HI, rß, we show that the operator LI, rß is conjugate to an integral operator KI, rß. If furthermore rß is real, then KI, rß is selfadjoint, so that LI, rß : HI, rß -> HI, rß has purely real spectrum. It is proved that LI, rß also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space C{omega} [0, 1], and on the Fréchet space C{infty} [0, 1]. The analytic properties of the map rß ↦ LI, rß are investigated. For certain alphabets I of an arithmetic nature (for example, I = primes, I = squares, I an arithmetic progression, I the set of sums of two squares it is shown that rß ↦ LI, rß admits an analytic continuation beyond the half-plane Re rß > {theta}I.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Faculty of Science > Mathematics |
| Library of Congress Subject Headings (LCSH): | Hyperbolic spaces, Geometry, Non-Euclidean, Transfer operators, Continued fractions, Ergodic theory |
| Journal or Publication Title: | Proceedings of the London Mathematical Society |
| Publisher: | Cambridge University Press |
| ISSN: | 0024-6115 |
| Date: | May 2003 |
| Volume: | Vol.86 |
| Number: | No.3 |
| Page Range: | pp. 755-778 |
| Identification Number: | 10.1112/S0024611502013904 |
| Status: | Peer Reviewed |
| Access rights to Published version: | Open Access |
| Funder: | Consejo Nacional de Ciencia y Tecnología (Mexico) [Mexican Council for Science and Technology] (CONACYT), National Science Foundation (U.S.) (NSF) |
| Grant number: | 110864/110990 (CONACYT), DMS 0100078 (NSF) |
| URI: | http://wrap.warwick.ac.uk/id/eprint/779 |
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