To appear in Israel Journal of Mathematics LYAPUNOV EXPONENTS FOR PRODUCTS OF MATRICES AND MULTIFRACTAL ANALYSIS. PART II: NON-NEGATIVE MATRICES
نویسنده
چکیده
We continue the study in [11, 14] on the upper Lyapunov exponents for products of matrices. Here the matrix function M(x) is only assumed to be non-negative. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. Anyway we focus our interest on a special case where M(x) takes finite values M1, . . ., Mm. In this case we prove the variational formula under an additional irreducibility condition. This extends a previous result of the author and Lau [14]. As an application, we prove a new multifractal formalism for a certain class of self-similar measures on R with overlaps. More precisely, let μ be the self-similar measure on R generated by a family of contractive similitudes {Sj = ρx + bj}j=1 which satisfies the finite type condition. Then we can construct a family (finite or countably infinite) of closed intervals {Ij}j∈Λ with disjoint interiors, such that μ is supported on ⋃ j∈Λ Ij and the restricted measure μ|Ij of μ on each interval Ij satisfies the complete multifractal formalism. Moreover the dimension spectrum dimH Eμ|Ij (α) is independent of j.
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