量子化一维映射半经典测度的熵界
Entropic bounds on semiclassical measures for quantized one-dimensional maps
摘要：
量子遍历性认为，一个量子化遍历系统的几乎所有本征态的无穷序列在相空间中是等分布的。另一方面，可能存在收敛于不同（非Liouville）经典不变测度的例外序列。利用N. Anantharaman和S. Nonnenmacher math-ph/0610019，arXiv：0704.1564（与H. Koch），对于Anosov测地线流，任何半经典测度的度量熵必有下界.结果似乎是最佳的一致膨胀系统，但不是在一般情况下，它可能会变得甚至平凡，如果曲率的黎曼流形是强非一致的。同样的作者已经证实，事实上，一个更强的界限（在一般情况下有效）应该成立。在目前的工作中，我们考虑这样的熵界使用的模型的量化一维地图。对于一类非一致扩张映射，证明了Anantharaman-Nonnenmacher猜想.此外，对于这些映射，我们能够构建一些显式序列的本征态饱和的界限。这表明，在这种情况下，约束边界实际上是最优的。
英文摘要：
Quantum ergodicity asserts that almost all infinite sequences of eigenstates of a quantized ergodic system are equidistributed in the phase space. On the other hand, there are might exist exceptional sequences which converge to different (non-Liouville) classical invariant measures. By the remarkable result of N. Anantharaman and S. Nonnenmacher math-ph/0610019, arXiv:0704.1564 (with H. Koch), for Anosov geodesic flows the metric entropy of any semiclassical measure must be bounded from below. The result seems to be optimal for uniformly expanding systems, but not in general case, where it might become even trivial if the curvature of the Riemannian manifold is strongly non-uniform. It has been conjectured by the same authors, that in fact, a stronger bound (valid in general case) should hold. In the present work we consider such entropic bounds using the model of quantized one-dimensional maps. For a certain class of non-uniformly expanding maps we prove Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case.
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