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Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices

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Abstract

We continue the study in [15, 18] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. In any case, we focus our interest on a special case where the matrix function M(x) takes finite values M 1, ..., M m . In this case, we prove the variational formula under an additional irreducibility condition. This extends a previous result of the author and Lau [18]. 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 {S j = ρx + b j } ℓ j=1 which satisfies the finite type condition. Then we can construct a family (finite or countably infinite) of closed intervals {I j } j∈Λ with disjoint interiors, such that μ is supported on ⋃ j∈Λ I j and the restricted measure \( \mu |_{I_j } \) of μ on each interval I j satisfies the complete multifractal formalism. Moreover, the dimension spectrum dim H \( E_{\mu |_{I_j } } \) (α) is independent of j.

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  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    De-Jun Feng

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Correspondence to De-Jun Feng.

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The author was partially supported by the direct grant and RGC grants (Projects 400706, 401008) in CUHK, Fok Ying Tong Education Foundation and NSFC (Grant 10571100).

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Feng, DJ. Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices. Isr. J. Math. 170, 355–394 (2009). https://doi.org/10.1007/s11856-009-0033-x

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  • DOI: https://doi.org/10.1007/s11856-009-0033-x

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