The resonance spectrum of the cusp map in the space of analytic functions

Details The resonance spectrum of the cusp map in the space of analytic functions The resonance spectrum of the cusp map in the space of analytic functions

2001-10-08

arxiv.org/abs/nlin/0110009v2

Nonlinear Sciences

Chaotic Dynamics

Submitted to JMP; The description of the spectrum in some Hardy spaces is added

Scientific paper

10.1063/1.1483895

We prove that the Frobenius--Perron operator $U$ of the cusp map $F:[-1,1]\to[-1,1]$, $F(x)=1-2\sqrt{|x|}$ (which is an approximation of the Poincar\'e section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that for any $q\in(0,1)$ the spectrum of $U$ in the Hardy space in the disk $\{z\in\C:|z-q|<1+q\}$ is the union of the segment $[0,1]$ and some finite or countably infinite set of isolated eigenvalues of finite multiplicity.

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