1996/10/07 by Hans Henrik Rugh, Rugh, Hans Henrik
Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #chao-dyn #nlin.CD
paper · pdf · doi:10.48550/arxiv.chao-dyn/9610011
22 pages, LaTeX
arxiv created 1996/10/07 · arxiv updated 2009/11/30
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) = σc ∪ σp where σc=[0,1] is the continuous spectrum of \cal 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 \cal 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.