On Transfer Operators for Continued Fractions with Restricted Digits

For I ⊂ N, let ΛI denote those numbers in the unit interval whose continued fraction digits all lie in I. Define the corresponding transfer operator LI,βf(z) = ∑ n∈I ( 1 n+z )2β f ( 1 n+z ) for Re(β) > max(0, θI), where Re(β) = θI is the abscissa of convergence of the series ∑ n∈I n −2β . When acting on a certain Hilbert space HI,β , we show that the operator LI,β is conjugate to an integral operator KI,β . If furthermore β is real, then KI,β is selfadjoint, so that LI,β : HI,β → HI,β has purely real spectrum. It is proved that LI,β also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space C[0, 1], and on the Fréchet space C∞[0, 1]. The analytic properties of the map β 7→ LI,β are investigated. For certain alphabets I of an arithmetic nature (eg. I = {primes}, I = {squares}, I an arithmetic progression, I the set of sums of two squares) it is shown that β 7→ LI,β admits an analytic continuation beyond the half-plane Re(β) > θI .