Brjuno functions and related topics

We investigate the fractal properties of the Brjuno-Yoccoz function Y associated with the nearest integer continued fraction. Similar to the conventional Brjuno function B, the Hölder exponents are not meaningful for Y. It is appropriate to replace the L∞ norm with an Lp norm and thus employ regularity in terms of membership in the spaces introduced by Calderón and Zygmund. We establish that the fractal properties of Y mirror those of B : the p-exponent at x is equal to the inverse of the irrationality exponent of x.