ON THE THERMODYNAMIC FORMALISM FOR THE GAUSS MAP

We study the generalized transfer operator ℒ β f(z)= {Mathematical expression} of the Gauss map Tx=(1/x) mod 1 on the unit interval. This operator, which for β=1 is the familiar Perron-Frobenius operator of T, can be defined for Re β>1/2 as a nuclear operator either on the Banach space A ∞(D) of holomorphic functions over a certain disc D or on the Hilbert space ℋ Reβ (2) (H -1/2 of functions belonging to some Hardy class of functions over the half plane H -1/2. The spectra of - β on the two spaces are identical. On the space ℋ Reβ (2) (H -1/2 ℒ β is isomorphic to an integral operator K β with kernel the Bessel function {Mathematical expression} and hence to some generalized Hankel transform. This shows that ℒ β has real spectrum for real β>1/2. On the space A ∞(D) the operator ℒ β can be analytically continued to the entire β-plane with simple poles at β=β k =(1-k)/2, k=0, 1, 2,..., and residue the rank 1 operator N (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 {Mathematical expression}, 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. © 1990 Springer-Verlag.