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Smoothness of the L q ‐Spectrum of Self‐Similar Measures with Overlaps
Author(s) -
Feng De-Jun
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070300437x
Subject(s) - mathematics , differentiable function , formalism (music) , spectrum (functional analysis) , real line , multifractal system , measure (data warehouse) , smoothness , probability measure , combinatorics , pure mathematics , mathematical analysis , discrete mathematics , physics , fractal , quantum mechanics , computer science , art , musical , database , visual arts
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.