On the thermodynamic formalism for the Gauss map

Abstract

We study the generalized transfer operator 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 of functions belonging to some Hardy class of functions over the half planeH −1/2. The spectra of on the two spaces are identical. On the space is isomorphic to an integral operator with kernel the Bessel function $$\mathfrak{F}_{2\beta - 1} (2\sqrt {st} )$$ and hence to some generalized Hankel transform. This shows that has real spectrum for real β>1/2. On the spaceA ∞(D) the operator can be analytically continued to the entire β-plane with simple poles at $$\beta = \beta _k = (1 - k)/2,k = 0,1,2,...$$ and residue the rank 1 operator $$\mathcal{N}^{(k)} f = \tfrac{1}{2}(1/k!)f^{(k)} (0)$$ . From this similar analyticity properties for the Fredholm determinant of and hence also for Ruelle's zeta function follow. Another application is to the function $$\zeta _M (\beta ) = \sum\limits_{n = 1}^\infty {[n]^\beta } $$ , where [n] denotes the irrational[n]=(n+(n 2+4)1/2)/2. ζM(β) extends to a meromorphic function in the β-plane with the only poles at β=±1 both with residue 1.