Counting periodic orbits on fractals weighted by their Lyapunov exponents

Several authors have shown that Kusuoka’s measure $\kappa$ on fractals is a scalar Gibbs measure; in particular, it maximises a pressure. There is also a different approach, in which one defines a matrix-valued Gibbs measure $\mu$ which induces both Kusuoka’s measure $\kappa$ 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 maximised by $\mu$. In the second part, we use the matrix-valued Gibbs measure $\mu$ to count periodic orbits on fractals, weighted by their Lyapounov exponents.