Intermittency and regularized Fredholm determinants

Abstract.

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 associated Perron-Frobenius operator ℳ has a continuous and residual spectrum contained in the line-segment σ _c =[0,1] and a point spectrum σ _p which has no points of accumulation outside 0 and 1. Furthermore, points in σ _p −{0,1} are eigenvalues of finite multiplicity. We construct a regularized Fredholm determinant d(λ) which has a holomorphic extension to λ∈ℂ−σ _c and can be analytically continued from each side of σ _c to an open neighborhood of σ _c −{0,1} (on different Riemann sheets). In ℂ−σ _c the zero-set of d(λ) is in one-to-one correspondence with the point spectrum of ℳ. Through the conformal transformation the function d∘λ(z) extends to a holomorphic function in a domain which contains the unit disc.