How (not) to prove (un)distortion for diffeomorphisms of one-manifolds

Abstract: This article addresses the following general question: Given a one-dimensional manifold $M$ and $1 \le r &lt; s \le \infty$, does there exist a $C<sup>s$ orientation preserving compactly supported diffeomorphism of $M$ that is undistorted in the group $\mathrm{Diff}<em>{c,+}<sup>s(M)$ of such diffeomorphisms while distorted in the bigger group of $C<sup>r$ diffeomorphisms? Interestingly, the answer is known to be positive in the case $(r,s)=(1,2)$ and negative in the case $(r,s)=(2,\infty)$, according to arXiv:2004.07055 and arXiv:2507.13770, respectively. The first part of this note originates from a failed attempt to extend the ideas of arXiv:2004.07055 to the case $(r,s)=(2,3)$. More precisely, in regularities $C<sup>1$ and $C<sup>2$, obstructions to distortion are provided by drifts of cocycles for isometric actions of $\mathrm{Diff}</em>{c,+}<sup>r(M)$ on Banach spaces for $r=1$ and $r=2$ (namely, the logarithmic and projective derivatives $f\mapsto \log Df$ and $f\mapsto D\log Df$, respectively). On $\mathrm{Diff}_{c,+}<sup>3(M)$, the so-called Liouville cocycle is a natural candidate when looking for new obstructions, but we show that its drift vanishes for $C<sup>2$-distorted diffeomorphisms (and this holds more generally for any "similar" cocycle). This does not rule out the existence of $C<sup>2$-distorted diffeomorphisms that are $C<sup>3$-undistorted. However, at least in the case of the real line, such a diffeomorphism should have very low regularity. Indeed, extending the methods and results of arXiv:2507.13770, in the second part of this article, we show that every compactly supported $C<sup>2$-distorted diffeomorphism of the real line is $C<sup>r$-distorted provided its differentiability class is larger than $C<sup>{2r+2}$.

• The paper shows that a vanishing Liouville cocycle drift characterizes when a C^2-diffeomorphism is almost-reducible or C^r-distorted.
• It leverages length functions and cocycle techniques from L^1 spaces to identify precise regularity thresholds and structural obstructions.
• The study reveals intrinsic regularity loss in almost-reducibility constructions, raising key open questions in one-dimensional dynamics.

(Un)distortion Phenomena in Groups of Diffeomorphisms of One-Manifolds: Length Functions, Cocycles, and Regularity

Introduction and Main Problem

This paper addresses the central question of when a diffeomorphism of a one-dimensional manifold, belonging to a higher regularity class $C^s$, can exhibit distortion in the group of $C^r$ diffeomorphisms ($r < s$), but not in the group of $C^s$ diffeomorphisms. The context is the group $Diff_{c,+}^r(M)$ of orientation-preserving, compactly supported $C^r$ diffeomorphisms of a one-manifold $M$. Here, an element is distorted if successive word lengths of its powers with respect to a finite generating set grow sublinearly, a concept with deep ties to geometric group theory and dynamics.

The known state of the art includes positive results for the pair $(r,s) = (1,2)$ [Na21] and negative results for $(r,s)=(2,\infty)$ [EBM25], implying that in the smooth ($C^\infty$) category, the obstructions to distortion and almost-reducibility are more severe. This work advances understanding at intermediate regularities, especially for $C^r$0, and systematically analyzes the relevant cocycles and their associated length functions.

Cocycles, Length Functions, and Drift: Structural Obstructions

One of the principal tools is the construction and analysis of length functions arising from cocycles for isometric group actions on Banach spaces, particularly $C^r$1 spaces. The approach generalizes the classical use of the logarithmic derivative in the $C^r$2 category (which yields obstructions via the drift associated to $C^r$3), to higher regularity settings.

Given a cocycle $C^r$4 for an isometric action $C^r$5 of a group $C^r$6 on a Banach space $C^r$7, the drift $C^r$8 for $C^r$9 is defined as the limiting averaged norm of $r < s$0. The vanishing of this drift provides a necessary condition for distortion.

A parade of cocycles is considered:

• $r < s$1 for $r < s$2,
• $r < s$3 (the projective derivative) for $r < s$4,
• The Liouville cocycle

$r < s$5

which encodes second-order geometry and connects to the Schwarzian derivative.

Key result: For $r < s$6-distorted diffeomorphisms, the drift associated to the Liouville cocycle vanishes, and more generally, no cocycle of the form considered (for the natural isometric action on $r < s$7, $r < s$8 arbitrary) can detect further obstructions for elements with vanishing asymptotic variation and no interior fixed points. This is formalized by a general vanishing theorem for such drifts.

There is a sharp characterization: vanishing drift for the Liouville cocycle occurs exactly when a diffeomorphism is either $r < s$9-conjugate to a restriction of a Möbius transformation or embeds in a $C^s$0 flow without hyperbolic fixed points; that is, the only obstruction comes from non-parabolic elements.

Asymptotic Variation, Mather Invariant, and Almost-Reducibility

The dynamical significance of the projective cocycle (and its corrected versions) is highlighted, relating the drift of $C^s$1 (called the asymptotic variation) to the variation of the Mather invariant. Explicitly, for $C^s$2 without interior fixed points, the paper refines prior inequalities to show

$C^s$3

with $C^s$4, providing a concrete measure of the obstruction for $C^s$5 to embed in a $C^s$6 flow.

Extensions are given to the context of cocycles on the circle, with the notable result that the Liouville drift vanishes for all $C^s$7 circle diffeomorphisms with irrational rotation number, reflecting the rigidity and abundance of almost-reducibility in this regime.

Quantitative Results and Regularity Loss in Almost-Reducibility

A strong theme is the relationship between almost-reducibility (the capacity to conjugate a diffeomorphism arbitrarily close to an isometry) and distortion. Using refined vector field interpolation and regularization, the results show that for $C^s$8 ($C^s$9) with vanishing asymptotic variation, $Diff_{c,+}^r(M)$0 is $Diff_{c,+}^r(M)$1-almost-reducible. Where $Diff_{c,+}^r(M)$2 fails to be $Diff_{c,+}^r(M)$3-flat, the regularity can be pushed up to $Diff_{c,+}^r(M)$4. Technical obstacles mean that in general, a loss of regularity is intrinsic to the argument, and eliminating this remains an open problem.

Strong Claim: If $Diff_{c,+}^r(M)$5 is $Diff_{c,+}^r(M)$6-almost-reducible, then $Diff_{c,+}^r(M)$7 is $Diff_{c,+}^r(M)$8-distorted. The proof exploits local perfectness properties in the group of compactly supported diffeomorphisms and advanced fragmentation and conjugacy techniques, involving a delicate analysis of commutator decompositions. This claims that, up to a loss of two derivatives, almost-reducibility guarantees distortion.

Implications, Open Questions, and Future Directions

These results build out a nuanced picture of the subgroup structure of diffeomorphism groups at varying regularity levels, identifying where cocycle drift invariants provide actual obstructions to distortion and almost-reducibility, and where their limitations lie.

Open directions include:

• Whether loss of regularity in almost-reducibility versus distortion is an artifact of current methods, or an unavoidable aspect for $Diff_{c,+}^r(M)$9.
• The construction of explicit diffeomorphisms at intermediate regularities ($C^r$0, $C^r$1) exhibiting the predicted phenomena.
• Developing alternative obstructions to distortion not expressible as drifts of cocycles for standard isometric actions, or realizing sharper controlling invariants.
• Exploring the upstream consequences for one-dimensional dynamics, rigidity problems, and geometric group theory.

The technical apparatus developed, especially in producing fine-tuned conjugacies and handling the regularization of vector fields, is likely to have further reach both in one-dimensional dynamics and in the (still less understood) study of higher-dimensional transformation groups.

Conclusion

This work establishes substantial new structure theorems for the distortion and (almost-)reducibility theory of diffeomorphism groups of one-manifolds at intermediate regularity. It demonstrates the essential role of cocycle-based length functions, quantifies the efficacy and sharpness of drift invariants, and clarifies the interplay between regularity, dynamical invariants, and group-theoretic properties. The negative results for cocycle drift obstructions above $C^r$2, and the precise regularity thresholds for almost-reducibility, delineate obstacles and avenues for future research in group actions and dynamics on one-manifolds.

Reference: "How (not) to prove (un)distortion for diffeomorphisms of one-manifolds" (2607.04001).