Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices

Abstract

Let (Σ,σ) be a full shift space on an alphabet consisting ofm symbols and letM: Σ→L ⁺(ℝ^d, ℝ^d) be a continuous function taking values in the set ofd×d positive matrices. Denote by λ _M (x) the upper Lyapunov exponent ofM atx. The set of possible Lyapunov exponents is just an interval. For any possible Lyapunov exponentα, we prove the following variational formula,$\begin{gathered} \dim \left\{ {x \in \Sigma :\lambda _M \left( x \right) = \alpha } \right\} = \frac{1}{{\log m}} _{q \in \mathbb{R}}^{\inf } \left\{ { - \alpha q + P_M (q)} \right\} \hfill \\ = \frac{1}{{\log m}} _\mu ^{\max } \left\{ {h\left( \mu \right):M_ * \left( \mu \right) = \alpha } \right\} \hfill \\ \end{gathered}$, where dim is the Hausdorff dimension or the packing dimension,P _M(q) is the pressure function ofM, μ is aσ-invariant Borel probability measure on Σ,h(μ) is the entropy ofμ, and$M_ * (\mu ) = \mathop {\lim }\limits_{x \to \infty } \frac{1}{n}\int {\log \left\| {M(y)M(\sigma y)...M(\sigma ^{n - 1} y)} \right\|d\mu \left( y \right)}$.