Intermittency and Regularized Fredholm Determinants

We consider real-analytic maps of the interval I = [0, 1] which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the spectrum of the associated Perron-Frobenius operator M has a decomposition sp(M) = σc ∪ σp where σc = [0, 1] is the continuous spectrum of M and σp is the pure point spectrum with no points of accumulation outside 0 and 1. We construct a regularized Fredholm determinant d(λ) which has a holomorphic extension to λ ∈ C − σc and can be analytically continued from each side of σc to an open neighborhood of σc − {0, 1} (on different Riemann sheets). In C − σc the zero-set of d(λ) is in one-to-one correspondence with the point spectrum of M. Through the conformal transformation λ(z) = 1 4z (1 + z) 2 the function d ◦ λ(z) extends to a holomorphic function in a domain which contains the unit disc. Shorttitle : Intermittency and Regularized Fredholm Determinants. 1 Assumptions and statement of results. Finding analytic continuations of functions, holomorphic in some a priori given domains, is a challenging and rewarding mathematical task in itself. In some cases it also provides elegant and non-trivial solutions to other problems in mathematics or physics. Thus, in dynamical systems theory the spectral properties of transfer operators and the zero sets of analytically extended holomorphic functions are related through Ruelle’s generalized Fredholm determinants and dynamical zeta functions. In establishing such a relationship