It is a known fact [1] that, for every $c\in (1,\infty]$, it is possible to find an alphabet $\mathcal{A}$ and a word $w\in \mathcal{A}^\omega$ such that $w$ has [critical exponent][4] $c$. It looks natural to me to define what I would call the *critical exponent function* in two steps as follows.  

**Step 1**

The $n$-*critical exponent* function $\kappa_n$:
\begin{equation}
x\in[0,1]\;\longmapsto \;\kappa_n(x)=\frac{1}{c_n(x)}
\end{equation}
where $c_n(x)$ is the critical exponent of the $n$-base expansion of $x$ (and it is intended $\frac{1}{\infty}=0$). Notice also that $\kappa_n$ is not affected by the possible ambiguity in the expansion of some rational points (the critical exponent is $\infty$ for both the possible expansions).

The range of $\kappa_n$ is $[0,\frac{4}{7}]$ for $n=3$, $[0,\frac{5}{7}]$ for $n=4$ and $\left[0,\frac{n-1}{n}\right]$ when $n=2$ or $n\ge 5$. This is due to a result by Rao [2] covering the last cases of a general conjecture by Dejean on repetition thresholds for finite alphabets [3]. 

**Step 2**

The *critical exponent* function $\kappa$:
\begin{equation}
\kappa: x\in[0,1]\;\longmapsto\; \sup_{n\in\mathbb{N}}{\kappa_n(x)}\in[0,1]
\end{equation}

It is easily seen that $\kappa(x)=0$ for every $x\in [0,1]$ such that its expansion in every base admits densities of the digits – in particular $\kappa$ vanishes on absolutely normal real numbers. Therefore, $\kappa$ is Lebesgue-measurable and $\int_0^1 \kappa(x)dx=0$. 

Looks like $\kappa$ has several unusual properties (it recalls loosely [Conway's base-13 function][5]). Almost every question about it I can think of seems non-trivial. I propose three of them.

**Q1**: Which Baire class (if any) does this function belong to?

**Q2**: Does $\kappa$ have fixed and/or periodic points (apart from the trivial fixed point 0)?

**Q3**: Does $\kappa$ attain the value 1?


[1]: Krieger, D., & Shallit, J. (2007). Every real number greater than 1 is a critical exponent. Theoretical computer science, 381(1-3), 177-182.

[2]: Rao, M. (2011). Last cases of Dejean's conjecture. Theoretical Computer Science, 412(27), 3010-3018.

[3]: Dejean, F. (1972). Sur un théorème de Thue. Journal of Combinatorial Theory, Series A, 13(1), 90-99.

[4]: https://en.wikipedia.org/wiki/Critical_exponent_of_a_word

[5]: https://en.wikipedia.org/wiki/Conway_base_13_function