On the thermodynamic formalism for the gauss map

Abstract

We study the generalized transfer operator ℒ _β f(z)=$\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{z + n}}} \right)^{2\beta } \times f\left( {1/\left( {z + n} \right)} \right)}$ of the Gauss mapTx=(1/x) mod 1 on the unit interval. This operator, which for β=1 is the familiar Perron-Frobenius operator ofT, can be defined for Re β>1/2 as a nuclear operator either on the Banach spaceA _∞(D) of holomorphic functions over a certain discD or on the Hilbert space ℋ ⁽²⁾_Reβ (H _-1/2 of functions belonging to some Hardy class of functions over the half planeH _−1/2. The spectra of − _β on the two spaces are identical. On the space ℋ ⁽²⁾_Reβ (H _-1/2 ℒ _β is isomorphic to an integral operatorK _β with kernel the Bessel function$\mathfrak{F}_{2\beta - 1} (2\sqrt {st} )$ and hence to some generalized Hankel transform. This shows that ℒ _β has real spectrum for real β>1/2. On the spaceA _∞(D) the operator ℒ _β can be analytically continued to the entire β-plane with simple poles atβ=β _k =(1-k)/2,k=0, 1, 2,..., and residue the rank 1 operatorN ^(k) f=1/2(1/K!)f ^(k)(0) . From this similar analyticity properties for the Fredholm determinant det (1-ℒ _β ) of ℒ _β and hence also for Ruelle's zeta function follow. Another application is to the function$\zeta _{\rm M} (\beta ) = \sum\limits_{n = 1}^\infty {\left[ n \right]^\beta }$, where [n] denotes the irradional [n]=(n+(n ²+4)^1/2)/2.ζ _M (β) extends to a meromorphic function in the β-plane with the only poles at β=±1 both with residue 1.