Abstract

Let $$\mu $$ be a self-similar measure generated by an IFS $$\varPhi =\{\phi _i\}_{i=1}^\ell $$ of similarities on $${{\mathbb {R}}}^d$$ ( $$d\ge 1$$ ). When $$\varPhi $$ is dimensional regular (see Definition 1.1), we give an explicit formula for the $$L^q$$ -spectrum $$\tau _\mu (q)$$ of $$\mu $$ over [0, 1], and show that $$\tau _\mu $$ is differentiable over (0, 1] and the multifractal formalism holds for $$\mu $$ at any $$\alpha \in [\tau _\mu '(1),\tau _\mu '(0+)]$$ . We also verify the validity of the multifractal formalism of $$\mu $$ over $$[\tau _\mu '(\infty ),\tau _\mu '(0+)]$$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to the recent result of Shmerkin (Ann. Math. (2) 189(2):319–391, 2019) on the $$L^q$$ -spectrum of self-similar measures.

Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.