This is a sharpening of the following problem: [$C^1$ function with a dense set of maximum values](https://mathoverflow.net/questions/392347/c1-function-with-a-dense-set-of-maximum-values).


**Problem set up:**


Let $f \colon [0, 1] \to \mathbb R$ be a function on the unit interval. We say $y \in \mathbb R$ is a local maximum value of $f$ if $y = f(x)$ for some strict local maximum $x$ of $f$.



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> **Question:** Let $r > 1$. Does there exist a function $f$ on the unit interval of bounded $r$-variation such that the set of local maximum values of $f$ is dense in some (nontrivial) open interval $(a, \, b)$?
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**Remarks:**


Note that a $C^0$ example is provided by a Brownian motion sample path, since a Brownian motion almost surely achieves a strict maximum in every interval. In the above problem, it is shown that a $C^1$ function with the above properties cannot exist, and I believe neither can a bounded variation one.


Thus an interesting question to ask is of bounded $r$-variation. The case $r = 1$ is impossible, and $r > 2$ is given by Brownian motion. For which $r$ precisely can this phenomenon occur?



Bounded $r$- variation function with a dense set of local maximum values
If you are happy with probabilistic constructions, fractional Brownian motion $X\_t^H$ gives an answer. If it's Hurst parameter is $H \in (0, 1)$, then the paths of $X\_t^H$ have infinite $1/H$-variation with probability one, and hence they necessarily attain local maxima in a dense set of $t$, just as the usual Brownian motion. On the other hand, the $r$-variation of $X\_t^H$ is finite with probability one for every $r > 1/H$, and of course such $r$ can be arbitrarily close to $1$ if one chooses $H$ appropriately.


A sample reference for the above is:


* Pratelli M. (2011) *A Remark on the $1/H$-Variation of the Fractional Brownian Motion*. In: Donati-Martin C., Lejay A., Rouault A. (eds) *Séminaire de Probabilités XLIII*. Lecture Notes in Mathematics, vol 2006. Springer, Berlin, Heidelberg. [DOI:10.1007/978-3-642-15217-7\_8](https://doi.org/10.1007/978-3-642-15217-7_8)


