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We study a new class of chaotic systems with dynamical localization, where gain or loss mechanisms break the Hermiticity, while allowing for parity-time (PT) symmetry. For a value gamma{PT} of the gain or loss parameter the spectrum undergoes a spontaneous phase transition from real (exact phase) to complex values (broken phase). We develop a one parameter scaling theory for gamma{PT}, and show that chaos assists the exact PT phase. Our results have applications to the design of optical elements with PT symmetry.

We study the Loschmidt echo F(t) for a class of dynamical systems showing critical chaos. Using a kicked rotor with singular potential as a prototype model, we found that the classical echo shows a gap (initial drop) 1-F_{g} , where F_{g} scales as F_{g}(alpha,,eta)=f_{cl}(chi_{cl} identical witheta;{3-alpha}/) ; alpha is the order of singularity of the potential, eta is the spread of the initial phase-space density, and is the perturbation strength. Instead, the quantum echo gap is insensitive to alpha , described by a scaling law F_{g}=f_{q}(chi_{q}=eta;{2}/) which can be captured by a random matrix theory modeling of critical systems. We trace this quantum-classical discrepancy to strong diffraction effects that dominate the dynamics.