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

We prove that the Frobenius-Perron operator $U$ of the cusp map $F:[-1,1]\to [-1,1]$, $F(x)=1-2 x^{1/2}$ (which is an approximation of the Poincare 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|