Several authors have shown that Kusuoka’s measure κ on fractals is a scalar Gibbs measure; in particular, it maximizes a pressure. There is also a different approach, in which one defines a matrix-valued Gibbs measure µ, which induces both Kusuoka’s measure κ and Kusuoka’s bilinear form. In the first part of the paper, we show that one can define a ‘pressure’ for matrix-valued measures; this pressure is maximized by µ. In the second part, we use the matrix-valued Gibbs measure µ to count periodic orbits on fractals, weighted by their Lyapounov exponents.

中文翻译：

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计算由 Lyapounov 指数加权的分形上的周期轨道

几位作者已经证明 Kusuoka 在分形上的测度κ是标量吉布斯测度；特别是，它使压力最大化。还有一种不同的方法，其中定义矩阵值吉布斯测度µ，从而导出 Kusuoka 测度κ和 Kusuoka 双线性形式。在本文的第一部分中，我们证明可以为矩阵值度量定义“压力”；该压力通过µ最大化。在第二部分中，我们使用矩阵值吉布斯测度µ来计算分形上的周期轨道，并由其 Lyapounov 指数加权。