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]→[−1,1], F(x)=1−2|x| (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∈(0,1) the spectrum of U in the Hardy space in the disk {z∈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.