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

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.

💡 Research Summary

The paper investigates the regularity of solutions to the twisted cohomological equation

  v(x) = α∘F(x) – (DF(x))^β α(x)

where F is an expanding C^{1+γ} map of the circle (not necessarily linear) and v is a very rough observable, possibly only a distribution. Classical results on the untwisted Livšic equation v = α∘F – α are well‑understood for Hölder data and linear expanding maps; however, the twisted case, especially with non‑linear F and non‑smooth v, has resisted analysis because transversality techniques require differentiability of v.

The authors introduce a novel tool: a fractional derivative operator D^β that is adapted to the dynamics of F. For the specific case F(x)=2x (mod 1) the operator satisfies natural chain and Leibniz rules

 D^β(θ∘F) = (D^βθ)∘F·(DF)^β, 
 D^β(θ·|Df|^β) = |Df|^β·D^βθ.

Applying D^β to the twisted equation yields

 D^βv·(DF)^β = (D^βα)∘F – D^βα,

which is precisely a standard Livšic equation for the unknown ψ = D^βα with observable φ = D^βv·(DF)^β. The observable φ is regular enough (typically Hölder) to invoke the classical Livšic theory: a continuous solution ψ exists if and only if the asymptotic variance σ²(φ) vanishes; otherwise only a distributional solution exists, and σ²(φ)>0 governs the statistical fluctuations of Birkhoff sums.

The paper’s main contributions are three families of dichotomy theorems.

Theorem A (Hölder data).
Let F be a C^{1+γ} expanding map of degree 2, β∈(0,γ), and v∈C^k with k≥γ. Define

 Ω_β = { v | the bounded solution α_{β,v} belongs to some C^{β+δ}, δ>0 }.

Then Ω_β is open and dense in C^k. For v∈Ω_β the solution α lies in C^β (indeed in C^γ) and is smooth; for v∉Ω_β the solution is nowhere differentiable, satisfies a β‑anti‑Hölder condition, and its graph has Hausdorff dimension > 1. Moreover, for analytic families t↦v_t the set of parameters where v_t∈Ω_β is either the whole interval or a countable set of isolated points, and the same holds for residual subsets of function spaces.

Theorem B (C^{1+γ} data).
If v∈C^{1+γ} and β∈(0,1), the set Λ = {β | v∈Ω_β} is either the whole interval (0,1) or a finite set. If the auxiliary function Dv + D²F·α_{v,1}·(DF)^{-1} is not cohomologous to a constant, Λ must be finite. In the former case α_{v,β} is C^{1+γ} for all β.

Theorem C (Besov data).
For v in the Besov space B^{γ}{1,1} (γ∈(0,1) and β∈(0,γ−½)) the same dichotomy holds: Ω_β is open and dense in B^{γ}{1,1}, the solution α belongs to B^{γ−δ}_{1,1} for any δ>0 when v∈Ω_β, and otherwise α is nowhere differentiable with the same anti‑Hölder and dimension properties. The theorem also includes analogous statements for analytic parameter families and residual subsets.

The proof strategy hinges on the fractional derivative reduction, the analysis of the variance σ²(φ) via thermodynamic formalism (transfer operators, pressure, and spectral gaps), and a careful study of distributional solutions of the Livšic equation. Sections 3–5 develop the chain and Leibniz rules for D^β, using Dirac approximations and connections to the fractional Laplacian. Sections 5–7 construct Birkhoff sum distributions and prove a Central Limit Theorem for them, showing that σ²(φ)>0 leads to Gaussian fluctuations and, consequently, to the irregularity of α. Section 9 demonstrates that σ²(φ) depends real‑analytically on parameters, which yields the parameter‑wise dichotomies.

The work extends earlier results that were limited to linear expanding maps (e.g., ℓx mod 1) and smooth observables. By allowing non‑linear expanding dynamics and observables as rough as distributions, the authors provide a unified framework that captures classical Brjuno, Weierstrass, and Takagi functions as particular solutions of the twisted equation. The dichotomy “smooth or nowhere differentiable” is shown to be a robust phenomenon, persisting under perturbations of both the map and the observable, and is intimately linked to the vanishing of the asymptotic variance in the associated Livšic problem. This bridges dynamical cohomology, fractal analysis, and probabilistic limit theorems, opening new avenues for studying irregular functions arising from dynamical systems.