Multifractal dimensions of product measures

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Abstract

We study the multifractal structure of product measures. For a Borel probability measure mu and q, t is an element of R, let H-mu(q,t) and P-mu(q,t) denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [011] Let mu be a Borel probability merasure on R(k) and let v be a Borel probability measure on R(l). Fix q, s, t is an element of R. We prove that there exists a number c > 0 such that

integral H-mu(q,s) (H-y) dH(nu)(q,t) (y) less than or equal to cH(mu x nu)(q, s+t) (H),

H-mu x nu(q, s+t) (E x F) less than or equal to cH(mu)(q,s) (E) P-nu(q,t) (F),

integral H-mu(q,s) (H-y) dP(nu)(q,t) (y) less than or equal to cP(mu x nu)(q, s+t) (H),

P-mu x nu(q, s+t) (E x F) less than or equal to cP(mu)(q, s) (E) P-nu(q, t) (F),

for E subset of or equal to R(k), F subset of or equal to R(l) and H subset of or equal to H subset of or equal to R(k+l) provided that mu and nu satisfy the so-called Pederer condition.

Using these inequalities we give upper and lower bounds for the multifractal spectrum of mu + nu in terms of the multifractal spectra of mu and nu.

Original languageEnglish
Pages (from-to)709-734
Number of pages26
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume120
Publication statusPublished - Nov 1996

Keywords

  • FRACTALS
  • DECOMPOSITIONS
  • SPECTRUM

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