Abstract
We exhibit scarring for the quantization of certain nonlinear ergodic maps on the torus. We consider perturbations of hyperbolic toral automorphisms preserving certain co-isotropic submanifolds. The classical dynamics is ergodic, hence, in the semiclassical limit almost all quantum eigenstates converge to the volume measure of the torus. Nevertheless, we show that for each of the invariant submanifolds, there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold.
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Communicated by P. Sarnak
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Kelmer, D. Scarring on Invariant Manifolds for Perturbed Quantized Hyperbolic Toral Automorphisms. Commun. Math. Phys. 276, 381–395 (2007). https://doi.org/10.1007/s00220-007-0331-2
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DOI: https://doi.org/10.1007/s00220-007-0331-2