[Paper Review] Distortion elements in $Diff^\infty(R/Z)$

Artur Avila|ArXiv.org|Aug 18, 2008

Geometric and Algebraic Topology5 references7 citations

TL;DR

This paper proves that every recurrent smooth diffeomorphism of the circle (in the sense of iterates approaching the identity) is a distortion element in the group $\mathrm{Diff}^\infty(\mathbb{R}/\mathbb{Z})$, meaning its iterates can be expressed as short compositions of finitely many generators. The key result resolves Franks and Handel's question by showing irrational rotations are distortion elements in the smooth category, using commutator constructions and rotation number theory.

ABSTRACT

We consider the group of smooth diffeomorphisms of the circle. We show that any recurrent $f$ (in the sense that $\{f^n\}_{n \in Z}$ is not discrete) is in fact a distortion element (in the sense that its iterates can be written as short compositions involving finitely many smooth diffeomorphisms). Thus rotations are distortion elements.

Motivation & Objective

• To resolve Franks and Handel's open question on whether irrational rotations are distortion elements in $\mathrm{Diff}^\infty(\mathbb{R}/\mathbb{Z})$.
• To establish that recurrence in the $C^\infty$ topology implies distortion in the group-theoretic sense.
• To construct short group-word expressions for elements close to the identity using commutators and compactly supported diffeomorphisms.
• To extend distortion theory to high regularity diffeomorphism groups, particularly for elements with irrational rotation numbers.
• To provide a framework applicable to higher-dimensional diffeomorphism groups, though not to volume-preserving or symplectic cases.

Proposed method

• Use of a metric $d_S$ on finitely generated subgroups to define distortion, requiring $d_S(f^n, \mathrm{id}) = o(n)$.
• Construction of sequences $\epsilon_n \to 0$ and $k_n \to \infty$ such that elements $h_n$ with $d(h_n, \mathrm{id}) < \epsilon_n$ satisfy $d_S(h_n, \mathrm{id}) \leq k_n$.
• Realization of elements near identity as products of commutators of diffeomorphisms supported on overlapping intervals, using a fixed finite set of generators.
• Application of Herman-Yoccoz theory on rotation numbers, particularly the existence of $C^\infty$ conjugacy to rotations for Diophantine rotation numbers.
• Use of a $C^\infty$ family of diffeomorphisms $\gamma_t$ with negative derivative to perturb identity and generate elements with Diophantine rotation numbers.
• Construction of a 1-parameter family of diffeomorphisms $h_t = [R_{1/2}f_tR_{1/2}, R_{1/2}g_tR_{1/2}][f_t,g_t]$ with positive derivative and nontrivial rotation number for $t>0$.

Experimental results

Research questions

• RQ1 Can irrational rotations be distortion elements in $\mathrm{Diff}^\infty(\mathbb{R}/\mathbb{Z})$?
• RQ2 Is every recurrent $C^\infty$ diffeomorphism of the circle a distortion element?
• RQ3 Can elements arbitrarily close to the identity be expressed as short products of commutators within a finitely generated subgroup?
• RQ4 What is the growth rate of distortion functions for smooth circle diffeomorphisms with irrational rotation numbers?
• RQ5 To what extent can the method of commutator decomposition be extended to higher-dimensional manifolds?

Key findings

• Any recurrent $f \in \mathrm{Diff}^\infty(\mathbb{R}/\mathbb{Z})$ is a distortion element, resolving Franks and Handel's question affirmatively in the smooth category.
• There exist sequences $\epsilon_n \to 0$ and $k_n \to \infty$ such that any $h_n$ with $d(h_n, \mathrm{id}) < \epsilon_n$ satisfies $d_S(h_n, \mathrm{id}) \leq k_n$ for some finitely generated subgroup $G$ with generating set $S$.
• Every countable set of recurrent elements in $\mathrm{Diff}^\infty(\mathbb{R}/\mathbb{Z})$ can be simultaneously made distortion elements in a single finitely generated subgroup.
• The construction ensures that for any function $r: \mathbb{N} \to \mathbb{N}$, each recurrent element can be made $r$-arbitrarily distorted, i.e., $d_S(f_i^m, \mathrm{id}) \leq n$ for some $m \geq r(n)$ and infinitely many $n$.
• The method relies on commutator decomposition and the existence of $C^\infty$ conjugacy to rotations for Diophantine rotation numbers, as per Herman-Yoccoz theory.
• The approach does not extend to real analytic diffeomorphisms, as the problem remains open in that category due to stronger algebraic rigidity.