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 spectrum of the associated Perron-Frobenius operator ${\cal M}$ has a decomposition $Sp ({\cal M}) = \sigma_c \cup \sigma_p$ where $\sigma_c=[0,1]$ is the continuous spectrum of ${\cal M}$ and $\sigma_p$ is the pure point spectrum with no points of accumulation outside 0 and 1. We construct a regularized Fredholm determinant $d(\lambda)$ which has a holomorphic extension to $\lambda \in C-\sigma_c$ and can be analytically continued from each side of $\sigma_c$ to an open neighborhood of $\sigma_c-{0,1}$ (on different Riemann sheets). In $C-\sigma_c$ the zero-set of $d(\lambda)$ is in one-to-one correspondence with the point spectrum of ${\cal M}$. Through the conformal transformation $\lambda(z) = 1/(4z) (1+z)^2$ the function $d \circ \lambda(z)$ extends to a holomorphic function in a domain which contains the unit disc.