On transfer operators for continued fractions with restricted digits

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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.

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