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Bessi, Ugo

Counting periodic orbits on fractals weighted by their Lyapounov exponents. (English) Zbl 07738145

Proc. Edinb. Math. Soc., II. Ser. 66, No. 3, 710-757 (2023).
Summary: Several authors have shown that Kusuoka’s measure \(\kappa\) 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 \(\mu \), 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 maximized 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.

MSC:

28A80 Fractals
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents

Keywords:

periodic orbits on fractals; Lyapunov exponents; matrix-valued Gibbs measures; Kusuoka’s measure
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References:

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