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

Stefano Marmi, Daniel Smania

TL;DR

This work analyzes twisted and Livšic cohomological equations for nonlinear expanding maps on the circle, introducing a fractional-derivative framework that links the two to reveal a Brjuno-like regularity dichotomy. By constructing a map $D^β$ adapted to the expanding map and leveraging Besov spaces, the authors show that the regularity of the twisted equation's solution is governed by the asymptotic variance $σ^2(φ)$, yielding either enhanced regularity or distributional behavior with Birkhoff sums and a Central Limit Theorem. The approach accommodates highly irregular data $v$ (even in Besov spaces with low regularity) and nonlinear dynamics, extending previous transversality-based results. The analysis yields real-analytic dependence on parameters in analytic families and provides a detailed toolkit of fractional-calculus rules, distributional solutions, and CLTs that tie together the twisted and Livšic equations in a unified regularity-dichotomy framework.

Abstract

Cohomological equations appear frequently in dynamical systems. One of the most classical examples is the Livšic equation $$ v(x) = α\circ F(x) - α(x).$$ The existence and regularity of its solutions $α$ is well understood when $F$ is a hyperbolic dynamical system (for instance an expanding map of the circle) and $v$ is a Hölder function. The $\textbf{twisted cohomological equation}$ $$ v(x) = α\circ F(x) - (DF(x))^β\, α(x) $$ 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 $α$ 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 Urbański, and more recently by Barański, Bárány 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 $α$ 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 Livšic cohomological equation, and to show that the resulting distributional solutions (in the sense of Schwartz) satisfy certain Central Limit Theorem.

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

TL;DR

This work analyzes twisted and Livšic cohomological equations for nonlinear expanding maps on the circle, introducing a fractional-derivative framework that links the two to reveal a Brjuno-like regularity dichotomy. By constructing a map adapted to the expanding map and leveraging Besov spaces, the authors show that the regularity of the twisted equation's solution is governed by the asymptotic variance , yielding either enhanced regularity or distributional behavior with Birkhoff sums and a Central Limit Theorem. The approach accommodates highly irregular data (even in Besov spaces with low regularity) and nonlinear dynamics, extending previous transversality-based results. The analysis yields real-analytic dependence on parameters in analytic families and provides a detailed toolkit of fractional-calculus rules, distributional solutions, and CLTs that tie together the twisted and Livšic equations in a unified regularity-dichotomy framework.

Abstract

Cohomological equations appear frequently in dynamical systems. One of the most classical examples is the Livšic equation The existence and regularity of its solutions is well understood when is a hyperbolic dynamical system (for instance an expanding map of the circle) and is a Hölder function. The 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 when is a linear endomorphism of the circle and is very regular. Notable contributions include works by Berry and Lewis; Ledrappier; Przytycki and Urbański, and more recently by Barański, Bárány 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 when is a expanding map of the circle and 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 Livšic cohomological equation, and to show that the resulting distributional solutions (in the sense of Schwartz) satisfy certain Central Limit Theorem.
Paper Structure (32 sections, 34 theorems, 321 equations)

This paper contains 32 sections, 34 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Twisted and Lvisic Cohomological equations
  3. Weierstrass, Takagi function and friends
  4. The Brjuno function
  5. Random Hölder Functions
  6. Main results
  7. Dichotomy for the regularity of Brujno-like functions
  8. Strategy of the proof
  9. Markovian expanding map
  10. Besov spaces
  11. Low regularity of the solutions of the twisted cohomologial equation
  12. Distributional solutions of the Livšic cohomological equation, asymptotic variance and the Central Limit Theorem
  13. The connection between the Livšic and twisted equations: fractional derivatives
  14. Dichotomies in real analytic families and Banach spaces of functions
  15. Chain and Leibniz rules for fractional derivatives
  16. ...and 17 more sections

Key Result

Theorem A

Let $F\colon \mathbb{S}^1 \rightarrow \mathbb{S}^1$ be a $C^{1+\gamma}$ expanding map on the circle with degree $2$ and $\gamma\in (0,1)$. Let $C^k(\mathbb{S}^1)$, with $k\geq \gamma$ be the Banach space of real-valued $C^k$ functions. For $\beta\in (0,\gamma)$ let Then $\alpha_{\beta,v} \in C^\beta(\mathbb{S}^1)$ for every $v\in C^k(\mathbb{S}^1)$. Moreover

Theorems & Definitions (76)

  • Theorem A: Dichotomy of Smoothness for Hölder data
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem B: Dichotomy of Smoothness for $C^{1+\gamma}$ data
  • Theorem C: Dichotomy of Smoothness for Besov data
  • Remark 2.4
  • Theorem 2.5: Marra, Morelli, and S. mms
  • Theorem D: Solutions with low regularity for Hölder input
  • Theorem E: Distributional solutions for distributional initial data
  • ...and 66 more