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

Abstract

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.