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 $\mathcal{M}$ has a decomposition $Sp(\mathcal{M})={\sigma }_c\cup {\sigma }_p$ where ${\sigma }_c=[0,1]$ is the continuous spectrum of $\mathcal{M}$ 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 $\mathcal{M}$. 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.

22 pages, LaTeX

Cited by in corpus (1)

• The role of singularities in chaotic spectroscopy