零温极限下具有两个极大值的势函数的测度选择
Selection of measures for a potential with two maxima at the zero temperature limit
摘要：
对于有限型$§=\{0,1,2\}^{\N}$的子移位，我们研究了仅允许两个极大化测度的非局部常数Hölder势的Gibbs测度在温度为零时的收敛性.这些测度是两个不同不动点上的狄拉克测度。在这两个不动点中的一个点上，电势是最平坦的。我们感兴趣的问题是：当温度趋于零时，不变的吉布斯态会选择这些概率中的哪一个？我们证明，一方面吉布斯措施收敛，并在另一方面，它不一定收敛到的措施，潜在的是最平坦的。我们考虑一个家庭的潜力，上述形式;其中一些有选择的凸组合的两个狄拉克措施，并为其他人有选择的狄拉克措施相关联的最平坦的点。在第一种情况下，如果我们考虑N.阿南塔拉曼河Iturriaga、P. Padilla和H.桑切斯·莫尔加多不变概率由本征函数和本征概率的结合得到。在我们的例子中发生的一个奇怪的现象是本征测度和本征函数具有相反的行为。当$β\to \infty$时，当与$1^\infty$周围的点相比时，特征度量在$0^\infty$周围成指数增大。对于本征函数，情况正好相反。
英文摘要：
For the subshift of finite type $§=\{0,1,2\}^{\N}$ we study the convergence at temperature zero of the Gibbs measure associated to a non-locally constant Hölder potential which admits only two maximizing measures. These measures are Dirac measures at two different fixed points. The potential is flattest at one of these two fixed points. The question we are interested is: which of these probabilities the invariant Gibbs state will select when temperature goes to zero? We prove that on the one hand the Gibbs measure converges, and at the other hand it does not necessarily converge to the measures where the potential is the flattest. We consider a family of potentials of the above form; for some of them there is the selection of a convex combination of the two Dirac measures, and for others there is a selection of the Dirac measure associated to the flattest point. In the first case this is contrary to what was expected if we consider the analogous problem in Aubry-Mather theory by N. Anantharaman, R. Iturriaga, P. Padilla and H. Sanchez-Morgado. The invariant probability is obtained by the junction of the eigen-function and the eigen-probability. A curious phenomena that happens in our examples is that the eigen-measure and the eigen-function have opposite behavior. When $β\to \infty$, the eigen-measure became exponential bigger around $0^\infty$, when compared to points around $1^\infty$. For the eigen-function the opposite happens.
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