A hypothesis about the average phase-space distribution of resonance eigenfunctions in chaotic systems with escape through an opening is proposed. Eigenfunctions with decay rate γ are described by a classical measure that (i) is conditionally invariant with classical decay rate γ and (ii) is uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle. This explains the localization of fast-decaying resonance eigenfunctions classically. It is found to occur in the phase-space region having the largest distance to the chaotic saddle. We discuss the dependence on the decay rate γ and the semiclassical limit. The hypothesis is numerically demonstrated for the standard map.

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Physical systems are often neither completely closed nor completely open, but instead they are best described by dynamical systems with partial escape or absorption. In this paper we introduce classical measures that explain the main properties of resonance eigenfunctions of chaotic quantum systems with partial escape. We construct a family of conditionally-invariant measures with varying decay rates by interpolating between the natural measures of the forward and backward dynamics. Numerical simulations in a representative system show that our classical measures correctly describe the main features of the quantum eigenfunctions: their multi-fractal phase space distribution, their product structure along stable/unstable directions, and their dependence on the decay rate. The (Jensen-Shannon) distance between classical and quantum measures goes to zero in the semiclassical limit for long-and short-lived eigenfunctions, while it remains finite for intermediate cases.

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