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Intermittency and regularized Fredholm determinants

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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 sigma(c) = [0, 1] and a point spectrum sigma(p) which has no points of accumulation outside 0 and 1. Furthermore, points in sigma(p) - {0, 1} are eigenvalues of finite multiplicity. We construct a regularized Fredholm determinant d(lambda) which has a holomorphic extension to lambda epsilon C - sigma(c) and can be analytically continued from each side of a, 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 M. Through the conformal transformation lambda(z) = 1/4z (1 + z)(2) the function d o lambda(z) extends to a holomorphic function in a domain which contains the unit disc

Topics: QA
Publisher: SPRINGER VERLAG
OAI identifier: oai:wrap.warwick.ac.uk:14979
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