Resonances of the Frobenius-Perron operator for a Hamiltonian map with a mixed phase space (Q2766233)

The authors consider a Hamiltonian system on a phase space NEWLINE\[NEWLINE(J_x,J_y,J_z)=(j\sin\theta\cos\varphi,j\sin\theta\sin\varphi,j\cos\theta).NEWLINE\]NEWLINE The dynamics is specified as a stroboscopic area preserving map \(M\) which consists of rotations \(R_z(\beta_z)\), \(R_y(\beta_y)\) about the \(y\) and \(z\)-axis, and a nonlinear rotation \(T_z(\tau)=R_z(\tau\cos\theta)\) about the \(z\)-axis which changes \(\varphi\) by \(\tau\cos\theta\), NEWLINE\[NEWLINEM=T_z(\tau)R_z(\beta_z)R_y(\beta_y),NEWLINE\]NEWLINE where they keep \(\beta_z=\beta_y=1\).NEWLINENEWLINENEWLINEThe Frobenius-Perron operator on the phase-space density is given by NEWLINE\[NEWLINEP\rho=\rho\circ M^{-1}.NEWLINE\]NEWLINE This operator is considered as an operator on \(L^2\), then it can be expressed as an infinite dimensional unitary matrix. They consider truncations \(P_n\) of this unitary matrix and study their eigenvalues. Some of these eigenvalues converge to the eigenvalues of \(P\) as \(n\) tends to infinity. But the rest of the eigenvalues remain in the unit circle. These will correspond to some of the eigenvalues of the Frobenius-Perron operator with domain bigger than \(L^2\). They study the properties of these eigenvalues and eigenfunctions, and also investigate the relations between these eigenvalues and the dynamical zeta function and the decay rate of correlation.