Smoothness of the L q ‐Spectrum of Self‐Similar Measures with Overlaps

Let μ be the self‐similar measure for a linear function system S j x =ρ x + b j ( j =1,2,…, m ) on the real line with the probability weight{ p j } j = 1 m . Under the condition that{ S j } j = 1 msatisfies the finite type condition, the L q ‐spectrum τ( q ) of μ is shown to be differentiable on (0,∞); as an application, μ is exact dimensional and satisfies the multifractal formalism.