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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. 755778. doi:10.1112/S0024611502013904

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Official URL: http://dx.doi.org/10.1112/S0024611502013904
Abstract
For any nonempty 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 halfplane 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, NonEuclidean, Transfer operators, Continued fractions, Ergodic theory  
Journal or Publication Title:  Proceedings of the London Mathematical Society  
Publisher:  Cambridge University Press  
ISSN:  00246115  
Official Date:  May 2003  
Dates: 


Volume:  Vol.86  
Number:  No.3  
Page Range:  pp. 755778  
DOI:  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) 
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