Multifractal Analysis and Dimension Spectrum of Unions

Question

Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets $$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) \,\mathrm{doesn't}\;\mathrm{exist}\right\}$$ where $d_\mu(x)$ denotes local dimension (or consider local entropy or Birkhoff Averages). We can easily say something about the dimension spectrum, namely $$\dim_H(\left\{x\mid d_\mu(x)=\alpha\right\}).$$

Can anything be said about $$\bigcup_{\alpha>0}\left\{x\mid d_\mu(x)=\alpha\right\}?$$ Moreover can we obtain the Hausdorff dimension of this (uncountable) union.

Answer 1

You are not detailed about the setting, but in many situations, say for an equilibrium measure of a Hölder continuous function on a repeller, the Hausdorff dimension of your union is the maximum (or sometimes the supremum) of the function $\alpha\mapsto\dim_H\left\{x: d_\mu(x)=\alpha\right\}$. This is due to the fact that the measure concentrated on the level set corresponding to the maximum can be used to obtain a lower bound for the dimension, which is the less immediate part of my claim. A similar statement holds for Gibbs measures on hyperbolic sets, although it is slightly more complicated since for most potentials defining the measure $\mu$ there will be no measure of maximal dimension. But still one can use certain noninvariant measures to estimate the dimension from below (notice that Frostman's lemma does not require invariant measures).