Kusuoka's measure on fractals is a Gibbs measure of a very special kind, since its potential is discontinuous while the standard theory of Gibbs measures requires continuous (in its simplest version, Hölder) potentials. In this paper we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar bilinear forms on fractals. Moreover, we shall see that Kusuoka's measure and bilinear form can be recovered in a simple way from the matrix-valued Gibbs measure.

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关于久保冈措施的另一种观点

Kusuoka的分形测度是一种非常特殊的Gibbs测度，因为它的势是不连续的，而Gibbs测度的标准理论需要连续的（最简单的形式是Hölder）势。在本文中，我们将看到，对于许多分形，完全可以在标准理论的范围内建立一类矩阵值的Gibbs测度。当然会有一些小的修改，但这仅仅是由于我们正在处理矩阵值函数和度量。我们将使用这些矩阵值的Gibbs测度在分形上建立自相似的双线性形式。此外，我们将看到，可以从矩阵值的吉布斯测度以简单的方式恢复Kusuoka测度和双线性形式。