In my paper with M. Hochman "Equidistribution from fractals", we give a geometric condition on a probability measure $\mu$ on the real numbers which ensures that $\mu$ almost every point is normal to a given base $m\ge 2$. The condition is fairly technical to state precisely, but it roughly says that if the measure $\mu$ does not exhibit any almost-periodic behaviour under the process of ``magnifying by a factor $m$ around a typical point'', then $\mu$ almost all points are normal to base $m$.

Our condition is invariant under $C^1$ maps. This does not say that normal numbers are invariant under $C^1$ maps, which is trivially false. It does say that some geometric features that ensure normality are, in a global sense, invariant under $C^1$ maps.

As pointed out by Robert Israel and Christian Remling, this is of interest only when $\mu$ is a singular measure with respect to Lebesgue measure. The good news it that there are plenty of natural singular measures to which our results apply. In particular, absolute continuity is definitely not the only reasonable condition that ensures that almost all points are normal (although it is the only condition if one considers ``size'' only).

For example, suppose $\mu$ is a measure on $[0,1)$ which is invariant under multiplication by $p$ on the circle, i.e. $\mu(A)=\mu(T_p^{-1}A)$ where $T_p(x)=px\bmod 1$. Then $\mu$ almost every point is normal to base $m$, for any $m$ such that $\log p/\log m$ is irrational (if $\log p/\log m$ is rational, this is not true). This extends previous results by Cassels, B. Host, E. Lindenstrauss and others.

For $T_p$-invariant measures $\mu$, it is not true that $\mu$ almost all points are normal to all bases. We do get many examples where this is the case.

Let $B\subset\mathbb{N}$ be a finite set with at least two elements, and let $A$ be the set of all points whose continued fraction expansion has only digits in the set $B$. It is well known that $A$ has positive and finite Hausdorff measure in its dimension, let $\mu$ be the restriction of the corresponding Hausdorff measure to $A$ (alternatively, there is a geometric Gibbs measure on $A$ for the Gauss map which is equivalent to $\mu$). Then $\mu$ almost all points are normal. Related results have been obtained by Kaufman, and by T. Jordan and T. Sahlsten.

To give a final example, let $A$ be the self-similar set obtained by replacing $[0,1]$ with $[0,1/2]\cup [2/3,1]$ and continuing inductively on each interval, always keeping intervals of relative lengths $1/2$ and $1/3$. Again, $A$ supports a natural measure $\mu$, and $\mu$ almost all points are normal, even though $A$ has Hausdorff dimension less than $1$. The same holds for a much wider class of self-similar measures, as long as there are two contraction ratios $r_1,r_2$ in the construction such that $\log r_1/\log r_2\notin\mathbb{Q}$.