Brjuno-Like Functions for nonlinear expanding maps: Fractional Derivatives and Regularity Dichotomies

TL;DR

This paper investigates the regularity of solutions to a twisted cohomological equation for nonlinear expanding maps on the circle, introducing fractional derivatives to analyze solutions with low regularity and establishing a Central Limit Theorem for their distributions.

Contribution

It develops a novel fractional derivative approach to study the regularity of solutions in nonlinear settings, extending understanding beyond linear cases.

Findings

Solutions can be distributional and satisfy a Central Limit Theorem.

Fractional derivatives effectively reduce the problem to classical Livšic equations.

Regularity properties depend on the nonlinear dynamics and the nature of the forcing function.

Abstract

Cohomological equations appear frequently in dynamical systems. One of the most classical examples is the Liv\v{s}ic equation $$ v(x) = \alpha \circ F(x) - \alpha(x).$$ The existence and regularity of its solutions $\alpha$ is well understood when $F$ is a hyperbolic dynamical system (for instance an expanding map of the circle) and $v$ is a H\"older function. The $\textbf{twisted cohomological equation}$ $$ v(x) = \alpha \circ F(x) - (DF(x))^\beta \, \alpha(x) $$ is much less well understood. Functions similar to the famous Brjuno, Weierstrass, and Takagi functions appear as solutions of this equation. This functional equation also appears in the work of M. Lyubich, and of Avila, Lyubich, and de Melo in their study of deformations of quadratic-like and real-analytic maps. Nevertheless, there are some striking results concerning the (lack of) regularity of solutions $\alpha$ when $F$ is a linear endomorphism of the circle and $v$ is very regular. Notable contributions include works by Berry and Lewis; Ledrappier; Przytycki and Urba\'nski, and more recently by Bara\'nski, B\'ar\'any and Romanowska, as well as by Shen, and by Ren and Shen, on Takagi and Weierstrass (and Weierstrass-like) functions. We study the regularity of solutions $\alpha$ when $F$ is a $\textbf{nonlinear}$ expanding map of the circle and $v$ is not differentiable or even continuous, a setting in which previously used transversality techniques do not appear to be applicable. The new approach uses fractional derivatives to reduce the study of the twisted cohomological equation to that of a corresponding Liv\v{s}ic cohomological equation, and to show that the resulting distributional solutions (in the sense of Schwartz) satisfy certain Central Limit Theorem.