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Journal Articles Inventiones Mathematicae Year : 2018

Multifractal analysis of the Brjuno function

Abstract

The Brjuno function B is a 1-periodic, nowhere locally bounded function, introduced by J.-C. Yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1. We show that T p α regularity, introduced by Calderón and Zygmund, is the only one which is relevant in order to unfold the pointwise regularity properties of B; we determine its T p α regularity at every point and show that it is directly related to the irrationality exponent τ (x): its p-exponent at x is exactly 1/τ (x). This new example of multifractal function puts into light a new link between dynamical systems and fractal geometry. Finally we also determine the Hölder exponent of a primitive of B.
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hal-04154230 , version 1 (06-07-2023)

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Stéphane Jaffard, Bruno Martin. Multifractal analysis of the Brjuno function. Inventiones Mathematicae, 2018, 212 (1), pp.109-132. ⟨10.1007/s00222-017-0763-z⟩. ⟨hal-04154230⟩
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