Entropy and dimension of disintegrations of stationary measures

Transactions of the American Mathematical Society, Series B Pub Date : 2019-08-05 DOI:10.1090/BTRAN/60

Pablo Lessa

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引用次数: 3

Abstract

We extend a result of Ledrappier, Hochman, and Solomyak on exact dimensionality of stationary measures for SL 2 ( R ) \text {SL}_2(\mathbb {R}) to disintegrations of stationary measures for GL ⁡ ( R d ) \operatorname {GL}(\mathbb {R}^d) onto the one dimensional foliations of the space of flags obtained by forgetting a single subspace.

The dimensions of these conditional measures are expressed in terms of the gap between consecutive Lyapunov exponents, and a certain entropy associated to the group action on the one dimensional foliation they are defined on. It is shown that the entropies thus defined are also related to simplicity of the Lyapunov spectrum for the given measure on GL ⁡ ( R d ) \operatorname {GL}(\mathbb {R}^d) .

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平稳测度分解的熵和维数

我们将Ledrappier, Hochman和Solomyak关于SL 2(R) \text {SL}_2(\mathbb {R})的平稳测度的精确维数的结果推广到GL (R d) \operatorname {GL}(\mathbb {R}^d)的平稳测度的分解到通过忽略单个子空间而得到的旗子空间的一维叶上。这些条件测度的维度用连续Lyapunov指数之间的间隙和与它们所定义的一维叶状上的群作用相关的一定熵来表示。结果表明，这样定义的熵也与给定测度GL (R d) \operatorname {GL}(\mathbb {R}^d)的李雅普诺夫谱的简单性有关。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Transactions of the American Mathematical Society, Series B

CiteScore

1.70

自引率

0.00%

发文量