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

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.

• The paper introduces fractional derivatives tailored to nonlinear expanding maps to reformulate twisted cohomological equations into Livšic-type equations.
• It demonstrates a dichotomy where generic solutions are nowhere differentiable with fractal properties, while exceptional cases gain full smoothness.
• The work leverages ergodic, thermodynamic, and wavelet methods to connect asymptotic variance with function regularity in dynamic systems.

Brjuno-Like Functions for Nonlinear Expanding Maps: Fractional Derivatives and Regularity Dichotomies

Introduction and Context

The paper "Brjuno-Like Functions for Nonlinear Expanding Maps: Fractional Derivatives and Regularity Dichotomies" (2601.15105) investigates the structure and regularity of solutions to twisted cohomological equations associated with expanding maps on the circle, focusing on their connection with classical objects such as the Brjuno, Weierstrass, and Takagi functions. These equations generalize the well-known Livšic equation, which is central in the study of dynamical systems, and offer a framework for understanding dichotomy phenomena in regularity—where solutions are either highly smooth or maximally irregular.

This work addresses cases that are not treatable by prior transversality techniques, especially when the map $F$ is nonlinear and observables $v$ may be singular or merely belong to rough function spaces (e.g., Besov spaces). The approach relies on constructing fractional derivatives tailored to the dynamics of $F$, connecting the twisted equation to a Livšic-type equation. It leverages ergodic and thermodynamical formalism for Markovian expanding maps and links the analysis to the Central Limit Theorem via asymptotic variances of associated observables.

Twisted Cohomological Equations and Brjuno-Like Functions

The main object of study is the twisted cohomological equation

$v(x) = \alpha(F(x)) - (DF(x))^\beta \alpha(x),$

where $F$ is a (piecewise) expanding map of $\mathbb{S}^1$ or an interval, $v$ is a function with possibly low regularity, and $\beta \in (0,1]$. The classical Livšic equation is a special case obtained by setting $\beta = 0$. This framework generalizes the functional equations satisfied by well-known nowhere differentiable functions like Weierstrass and Takagi, whose regularity and fractal properties are deeply tied to dynamics.

A salient feature of these equations is a regularity dichotomy for solutions: either they are very smooth (e.g., H\"older or even analytic), or they are highly irregular, with properties such as nowhere differentiability and increased Hausdorff dimension of their graphs, akin to random H\"older functions.

Framework and Techniques

Markovian Expanding Maps and Besov Spaces

The analysis is performed in the context of Markovian expanding (possibly nonlinear) maps $F$ admitting a symbolic Markov partition. Function spaces are defined via these partitions, including Besov spaces $\mathcal{B}^s_{1,1}$ and $\mathcal{B}^s_{\infty,\infty}$, providing sharp control over fine regularity properties. These spaces admit unconditional wavelet bases, enabling distributional analysis when $v$ is only in $L^1$ or lower-regularity Besov classes.

Fractional Derivatives

Central to the paper is a fractional differentiation operator $D^\beta$ designed to respect the dynamics of $F$, and satisfying (up to regularizing corrections) chain and Leibniz rules: $$D^\beta (\theta \circ F) \approx (D^\beta \theta)\circ F \cdot (DF)^\beta,$$

$$D^\beta (\theta \cdot (DF)^\beta) \approx (DF)^\beta D^\beta \theta.$$

This allows recasting the regularity problem for the twisted equation as one for a Livšic equation with fractional derivatives of the data.

Reduction to Livšic Equations, Asymptotic Variance, CLT

By differentiating the twisted equation, the smoothness of solutions $\alpha$ is mirrored by the existence of regular solutions to an associated Livšic equation: $\phi = \psi \circ F - \psi$ with

$$\phi = \frac{D^\beta v + \text{correction terms}}{(DF)^\beta}.$$

For typical $v$, solutions $\psi$ exist if and only if the asymptotic variance

$$\sigma^2(\phi) = \lim_n \int \left(\frac{1}{\sqrt{n}}\sum_{k < n} \phi\circ F^k\right)^2 d\mu$$

vanishes. Otherwise, the formal solution is a divergent series interpreted as a Birkhoff sum representing a distribution, and the wild behavior is quantified via the Central Limit Theorem.

Main Results: Regularity Dichotomies

The key theorem asserts that for a $C^{1+\gamma}$ expanding map $F$ of degree $2$ and suitably regular $v$, the set $\Omega_\beta$ of $v$ for which the solution $\alpha$ is not smoother than $C^\beta$ is open and dense in the relevant function space. More precisely:

• For generic data $v$ (open and dense, and prevalent in measure-theoretic senses for parametric families), the solution $\alpha_{\beta,v}$ is nowhere differentiable, satisfies an anti-H\"older property, and its graph has Hausdorff dimension greater than one (fractal-like).
• For exceptional $v$, $\alpha_{\beta,v}$ gains full smoothness. The criterion is the vanishing of the asymptotic variance of the associated observable after fractional differentiation.
• In analytic parametric families, non-generic (smooth) behavior occurs only at isolated (countable) parameter values.
• Similar dichotomies hold when varying $\beta$ or working in rougher function classes (e.g., Besov spaces $B^\gamma_{1,1}$), provided $\gamma - \beta > 1/2$.

These results generalize and unify phenomena observed for the Weierstrass and Takagi functions and the Brjuno function, and extend them to nonlinear expanding dynamics, previously inaccessible due to technical obstacles.

Numerical results include explicit bounds for regularity losses, and the demonstration that for generic $v$, the modulus of continuity of solutions is random-like (exhibiting a Central Limit Theorem for local averages).

Theoretical and Practical Implications

The established framework provides a complete characterization of the regularity dichotomy for solutions to twisted cohomological equations over nonlinear expanding maps. The fractional derivative constructions and thermodynamic formalism bridge smooth and distributional regimes, allowing the transfer of classical harmonic analysis techniques to the setting of dynamical systems and symbolic Markov partitions.

Practically, these results equip researchers with tools to describe the typical behavior of coboundary-type functions arising in deformations of dynamical systems, small divisor problems, and spectral theory (e.g., quantum modular forms, analytic number theory). The descriptions of function spaces and operator spectra are directly applicable to algorithms for computing invariant measures, fractal dimensions, and limit distributions for chaotic maps.

Theoretically, the link between variance vanishing and smoothness opens avenues for deeper investigation of dynamical dichotomies, especially in long-standing problems concerning the dimension and modulus of continuity of fractal functions defined by iterated systems or random processes.

Prospects for Future Research

Future work may focus on:

• Extending fractional differentiation and dichotomy results to more general dynamical systems (e.g., systems with multiple branches, higher dimensions).
• Investigating connections with quantum modular forms and the Nyman-Beurling criterion in analytic number theory.
• Exploring numerical methods to approximate asymptotic variances and distributions of Birkhoff sums for rough observables.
• Studying stochastic versions and random dynamical systems, particularly those modeling random H\"older functions and their dimensions.
• Addressing cases (such as the Gauss map for the Brjuno function) which require further development in the theory of distributional solutions and transfer operators.

Conclusion

The paper provides a rigorous and unified account of the regularity dichotomies exhibited by solutions to twisted cohomological equations for nonlinear expanding maps, relying on fractional derivative constructions matched to the dynamics. It establishes structural results concerning prevalence, openness, and measure-theoretic genericity of wild behavior, solidly connecting dynamical systems, harmonic analysis, ergodic theory, and fractal geometry. The implications span both foundational theory and computational practice, reinforcing the significance of functional-analytic and probabilistic perspectives in modern dynamical systems research.