On self-affine measures with equal Hausdorff and Lyapunov dimensions

2015/11/21 by Rapaport, Ariel · 1 citation
#28A80 (Secondary) #37C45 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1511.06893

Abstract

Let μ be a self-affine measure on ℝ^d associated to a self-affine IFS \φ_λ(x) = A_λx + v_λ\_λ∈Λ and a probability vector p=(p_λ)_λ>0. Assume the strong separation condition holds. Let γ₁≥...≥γ_d and D be the Lyapunov exponents and dimension corresponding to \A_λ\_λ∈Λ and p^ℕ, and let G be the group generated by \A_λ\_λ∈Λ. We show that if γ_m+1>γ_m=...=γ_d, if G acts irreducibly on the vector space of alternating m-forms, and if the Furstenberg measure μ_F satisfies dim_Hμ_F+D>(m+1)(d-m), then μ is exact dimensional with dimμ=D.

Cited by

• Hausdorff dimension of the Wedding cake type surfaces

Related

• On self-affine measures associated to strongly irreducible and proximal systems
• Hausdorff Dimension of Planar Self-Affine Sets and Measures with Overlaps
• Hausdorff dimension of planar self-affine sets and measures
• Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure
• Dimension of diagonal self-affine measures with exponentially separated projections