Slow and fast convergence to local dimensions of self‐similar measures

Let K and μ be the self‐similar set and the self‐similar measure associated with an iterated function system with probabilities ( S i , p i ) i =1,…, N satisfying the Open Set Condition. Let Σ = {1, …, N } ℕ denote the full shift space and let π : Σ → K denote the natural projection. The (symbolic) local dimension of μ at ω ∈ Σ is defined by $ \lim _{n} { {\log \mu K _{\omega \vert n}} \over {\log {\rm diam} \, K_{\omega \vert n} } } $ where K ω | n = S   ω   1○ … ○ S   ω   n( K ) for ω = ω 1 ω 2 … ∈ Σ, and the (symbolic) multifractal spectrum of μ is defined by\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document} $$ f_{s} (\alpha) := {\rm dim} \, \pi \,\bigg \{ \omega \in \Sigma \; \bigg \vert \lim _n \, { {\log \mu K _{\omega \vert n}} \over {\log {\rm diam} \, K_{\omega \vert n} } }= \alpha \bigg \} \, , \quad \alpha \ge 0 \, , $$ \end{document}where dim denotes the Hausdorff dimension. In this paper we study the speed with which the ratio $ { {\log \mu K _{\omega \vert n}} \over {\log {\rm diam} \, K_{\omega \vert n} } } $ converges. In particular, we prove that for all (sufficiently large) speeds γ , the set of points\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document} $$ \bigg \{ \omega \in \Sigma \; \bigg \vert \limsup _n \; { {\bigg \vert \log \mu K _{\omega \vert n} - \alpha \log {\rm diam} \, K _{\omega \vert n} \bigg \vert} \over {\sqrt {n \log \, \log n} } } = \gamma \bigg \} $$ \end{document}for which the ratio log $ { {\log \mu K _{\omega \vert n}} \over {\log {\rm diam} \, K_{\omega \vert n} } } $ converges to its limit with speed equal to γ , has full dimension. i.e.\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document} $$ {\rm dim} \; \pi \bigg \{ \omega \in \Sigma \; \bigg \vert \limsup _n \; { {\bigg \vert \log \mu K _{\omega \vert n} - \alpha \log {\rm diam} \, K _{\omega \vert n} \bigg \vert} \over {\sqrt {n \log \, \log n} } } = \gamma \bigg \} = f_{s} (\alpha) \, . $$ \end{document}(© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)