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## Studia Mathematica

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## A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Möbius function

### Tom 251 / 2020

Studia Mathematica 251 (2020), 195-206 MSC: Primary 37B05; Secondary 37B10, 37A35, 11Y35. DOI: 10.4064/sm180719-13-12 Opublikowany online: 23 August 2019

#### Streszczenie

A recent result of Downarowicz and Serafin (2019) shows that there exist positive entropy subshifts satisfying the assertion of Sarnak’s conjecture. More precisely, it is proved that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression (the Möbius function is an example of such a sequence $y$) then for every $N\ge 2$ there exists a subshift $\varSigma$ over $N$ symbols, with entropy arbitrarily close to $\log N$, uncorrelated to $y$.

In the present note, we improve the above result. First of all, we observe that the uncorrelation is uniform, i.e., for any continuous function $f:\varSigma \to \mathbb R$ and every $\epsilon \gt 0$ there exists $n_0$ such that for any $n\ge n_0$ and any $x\in \varSigma$ we have $$\bigg |\frac 1n\sum _{i=1}^{n}f(T^ix)\,y_i\bigg | \lt \epsilon .$$ More importantly, by a fine-tuned modification of the construction we create a strictly ergodic subshift with the desired properties (uniformly uncorrelated to $y$ and with entropy arbitrarily close to $\log N$).

The question about these two additional properties (uniformity of uncorrelation and strict ergodicity) has been posed by Mariusz Lemańczyk in the context of the so-called strong MOMO (Möbius Orthogonality on Moving Orbits) property. Our result shows, among other things, that strong MOMO is essentially stronger than uniform uncorrelation, even for strictly ergodic systems.

#### Autorzy

• Tomasz DownarowiczFaculty of Pure and Applied Mathematics
Wrocław University of Science and Technology
Wybrzeże Wyspiańskiego 27
50-370 Wrocław, Poland
e-mail
• Jacek SerafinFaculty of Pure and Applied Mathematics
Wrocław University of Science and Technology
Wybrzeże Wyspiańskiego 27
50-370 Wrocław, Poland
e-mail

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