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Several authors have shown that Kusuoka's measure $\kappa$ on fractals is ascalar Gibbs measure; in particular, it maximises a pressure. There is also adifferent approach, in which one defines a matrix-valued Gibbs measure $\mu$which induces both Kusuoka's measure $\kappa$ and Kusuoka's bilinear form. Inthe first part of the paper we show that one can define a "pressure" for matrixvalued measures; this pressure is maximised by $\mu$. In the second part, weuse the matrix-valued Gibbs measure $\mu$ to count periodic orbits on fractals,weighted by their Lyapounov exponents.

Counting periodic orbits on fractals weighted by their Lyapunov exponents