Journal de l'École polytechnique
Mathématiques

Alexander I. BUFETOV & Boris SOLOMYAK

Corrigendum to "Hölder regularity for the spectrum of translation flows"

Tome 9 (2022), p. 1513-1514.

https://doi.org/10.5802/jep.212

© Les auteurs, 2022.

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Publié avec le soutien
du Centre National de la Recherche Scientifique
CORRIGENDUM TO

“HÖLDER REGULARITY FOR THE SPECTRUM OF
TRANSLATION FLOWS”

BY ALEXANDER I. BUFETOV & BORIS SOLOMYAK

ABSTRACT. — The paper corrects the formulation of [3, Prop. 5.4], whose proof is unchanged.

RÉSUMÉ (Correction à “Régularité Hölder pour le spectre des flots de translation” )

Cet article corrige la formulation de [3, Prop. 5.4], dont la preuve est inchangée.

In [3, Prop. 5.4] a different order of quantifiers was needed from the one stated.
The following proposition was, in fact, proven in the paper, and this is the correct
formulation:

PROPOSITION. — For any $\varepsilon_1 > 0$ there exist $\delta_0 > 0$ and $\varrho > 0$ such that for $\mathbb{P}$-a.e. $\mathbf{a} \in \Omega_q$ and any $\delta \in (0, \delta_0)$, for all $B > 1$, and every Oseledets subspace $H_{\mathbf{a}}^J$ corresponding to $\mathbf{a}$, containing the unstable subspace $E_{\mathbf{a}}^u$,

$$
\dim_H(\mathcal{E}(\varrho, \delta, B) \cap H_{\mathbf{a}}^J) \le \dim(H_{\mathbf{a}}^J) - \kappa + \varepsilon_1,
$$

where $\kappa = \dim(E_{\alpha}^{u})$.

The short derivation of [3, Th. 4.1] from [3, Prop. 5.4] in this form and the proof of
the proposition do not require any changes.

Incidentally, exactly the same mixed-up parameter dependence is present in
[2, Prop. 9.1], and the correction is also the same. We would like to note that the
proposition with the order of quantifiers as it was published, is, in fact, correct when
the Rauzy induction is periodic; essentially, in the case of a single substitution, as it
was shown in [1].

*Acknowledgements*. — We are grateful to Rodrigo Treviño who pointed out the mistake to us.

MATHEMATICAL SUBJECT CLASSIFICATION (2020). — 37A30, 37B1, 37E35, 28A78.

KEYWORDS. — Translation flows, spectral measures, matrix Riesz products, upper Lyapunov exponents, Erdős-Kahane argument, Bratteli-Vershik automorphisms, renormalization cocycle.
REFERENCES

[1] A. I. BUFETOV & B. SOLOMYAK – “On the modulus of continuity for spectral measures in substitution dynamics”, *Adv. Math.* **260** (2014), p. 84–129.

[2] —, “The Hölder property for the spectrum of translation flows in genus two”, *Israel J. Math.* **223** (2018), no. 1, p. 205–259.

[3] —, “On the modulus of continuity for spectral measures in substitution dynamics”, *J. Éc. polytech. Math.* **8** (2021), p. 279–310.

Manuscript received 31st May 2022
accepted 26th September 2022

ALEXANDER I. BUFETOV, Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373,
39 rue F. Joliot Curie, Marseille, France
Steklov Mathematical Institute of RAS, Moscow, Russia
Institute for Information Transmission Problems, Moscow, Russia
E-mail: bufetov@mi.ras.ru

BORIS SOLOMYAK, Department of Mathematics, Bar-Ilan University
Ramat-Gan, Israel
E-mail: bsolom3@gmail.com
Url: https://u.math.biu.ac.il/~solomyb/personal.html