On the resonance eigenstates of an open quantum baker map

We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, |z_min| ⩽ |z| ⩽ |z_max|. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius r. We prove that if the moduli converge to r = |z_max| then the sequence of eigenstates is associated with a fixed phase space measure ρ_max. The same holds for sequences with eigenvalue moduli converging to |z_min|, with a different limit measure ρ_min. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius |z_min| < r < |z_max| there is no unique limit measure, and we identify some families of eigenstates with precise self-similar properties.