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首页> 外文期刊>Discrete and continuous dynamical systems >APPROXIMATIONS FOR GIBBS STATES OF ARBITRARY HOLDER POTENTIALS ON HYPERBOLIC FOLDED SETS
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APPROXIMATIONS FOR GIBBS STATES OF ARBITRARY HOLDER POTENTIALS ON HYPERBOLIC FOLDED SETS

机译：双曲折叠集上的任意持有者势的吉布斯状态逼近

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摘要

We give approximations for the Gibbs states of arbitrary Holder potentials φ, with the help of weighted sums of atomic measures on preimage sets, in the case of smooth non-invertible maps hyperbolic on folded basic sets A. The endomorphism may have also stable directions on A and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (instead they depend on the whole past and may intersect each other both inside and outside A). We consider here simultaneously all n-preimages in A of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set A. In particular we obtain the distribution of preimage sets on A, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit A, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in A. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an inverse SRB measure, for Lebesgue-almost all points.
机译：在光滑的不可逆映象在折叠的基本集合A上双曲的情况下，借助于原像集上原子度量的加权和，我们可以给出任意Holder势φ的吉布斯状态的近似值。 A和通常是不可扩展的。相空间的折叠意味着我们没有局部不稳定歧管的叶状结构（相反，它们取决于整个过去，并且可能在A内外彼此相交）。在此，我们同时考虑了点A中的所有n个原像，而不是通常的方法，即仅从某个给定的历史记录中获取连续的原像。因此，相对于鞍型折叠集A上的各种平衡度量，我们获得了连续的原像集的加权分布。特别是，相对于最大熵的度量，我们获得了A上原像集的分布。我们的结果不是Birkhoff遍历定理在逆极限A上的直接应用，因为一个点的史前集合通常是不可数的，并且对于A中的不同史前集合，收敛速度可能会有所不同。对于双曲环形内同形，我们获得对于Lebesgue几乎所有点，连续原像的分布都朝着SRB逆度量。

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