Distortion element in group of diffeomorphisms of the 2-sphere

Distortion in Groups of Circle and Surface Diffeomorphisms

In these lectures we consider how algebraic properties of discrete subgroups of Lie groups restrict the possible actions of those groups on surfaces. The results show a strong parallel between the possible actions of such a group on the circle $S^1$ and the measure preserving actions on surfaces. Our aim is the study of the (non)-existence of actions of lattices in a large class of non-compact Lie groups on surfaces. A definitive analysis of the analogous question for actions on $S^1$ was carried out by É. Ghys. Our approach is topological and insofar as possible we try to isolate properties o…

Maximal pseudometrics and distortion of circle diffeomorphisms

We initiate a study of distortion elements in the Polish groups $\mbox{Diff}_+^k(\mathbb{S}^1)$ ($1\leq k<\infty$), as well as $\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$, in terms of maximal metrics on these groups. We classify distortion in the $k=1$ case: a $C^1$ circle diffeomorphism is $C^1$-undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms with only non-hyperbolic fixed points which are $C^{1+AC}$-undistorted, and hence $C^k$-undistorted for all $k\geq 2$. In the appendix, we exhibit a maxi…

On proper extensions of the conformal group of sphere diffeomorphisms

In this paper, we prove that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. In addition, we show that any such group must always contain an element of positive topological entropy. We also provide an elementary characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that such a diffeomorphism of the interval with no fixed point in the interior contains the identity in the closure of its C^{1+bv} conjugacy class if and only if it is the time-1 map of a C^1 vector field. A corollary of this is that diffeomorphisms that do not arise from vector fields are undistorted in the whole group of interval interval diffeom…

Diffeomorphism groups of balls and spheres

In this paper we discuss the relationship between groups of diffeomorphisms of spheres and balls. We survey results of a topological nature and then address the relationship as abstract (discrete) groups. We prove that the identity component Diff_0(S^{2n-1}) of the group of smooth diffeomorphisms of S^{2n+1} admits no nontrivial homomorphisms to the group of C^1 diffeomorphisms of the ball B^m for any n and m. This result generalizes theorems of Ghys and Herman. We also examine finitely generated subgroups of Diff_0(S^n) and produce an example of a finitely generated torsion free group Gamma w…

Distortion elements in $Diff^\infty(R/Z)$

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.

Periods of Morse--Smale diffeomorphisms on $\mathbb{S}^n$, $\mathbb{S}^m \times \mathbb{S}^n$, $\mathbb{C}P^n}$ and $\mathbb{H}P^n}$

We study the set of periods of the Morse--Smale diffeomorphisms on the $n$-dimensional sphere $\mathbb{S}^n$, on products of two spheres of arbitrary dimension $\mathbb{S}^m \times \mathbb{S}^n$ with $m \neq n$, on the $n$-dimensional complex projective space $\mathbb{C}\text{\textbf{P}}^n}$ and on the $n$-dimensional quaternion projective space $\mathbb{H}\text{\textbf{P}}^n}$. We classify the minimal sets of Lefschetz periods for such Morse--Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function.

Distortion in the group of circle homeomorphisms

Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.

Representations of some lattices into the group of analytic diffeomorphisms of the sphere $\mathbb{S}^2$

In \cite{Ghys} it is proved that any morphism from a subgroup of finite index of $\mathrm{SL}(n,\mathbb{Z})$ to the group of analytic diffeomorphisms of $\mathbb{S}^2$ has a finite image as soon as $n\geq 5$. The case $n=4$ is also claimed to follow along the same arguments; in fact this is not straightforward and this case indeed needs a modification of the argument. In this paper we recall the strategy for $n\geq 5$ and then focus on the case $n=4$.

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…

éléments de distorsion du groupe des difféomorphismes isotopes à l'identité d'une variété compacte

We consider, on a compact manifold, the group of diffeomorphisms that are isotopic to the identity. We show that every recurrent element is a distorsion element. This generalizes Avila's theorem on circle diffeomorphisms. The method also provides a new proof of a result by Calegari and Freedman: on a sphere, in the group of homeomorphisms that are isotopic to the identity, every element is distorted.

Distortion Elements in Group actions on surfaces

If $\G$ is a finitely generated group with generators $\{g_1,...,g_j\}$ then an infinite order element $f \in \G$ is a {\em distortion element} of $\G$ provided $\displaystyle{\liminf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface and let $\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result shows that if $S$ has genus at least two and if $f$ is a distortion element in some finitely generated subgroup of $\Diff(S)_0$, then $\supp(μ) \subset \Fix(f)$ for every $f…

如何（不）证明一维流形上微分同胚的（非）扭曲性

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

研究给定一维流形\(M\)及\(1\leq r\lt s\leq\infty\)，是否存在\(C^s\)保向紧支微分同胚在\(\mathrm{Diff}_{c,+}^s(M)\)中无扭曲但在更大的\(C^r\)微分同胚群中有扭曲。分析了不同\((r,s)\)情况及相关阻碍。

Des points fixes communs pour des difféomorphismes de S^2 qui commutent et préservent une mesure de probabilité

We prove the existence of common fixed points for commuting homeomorphisms of the plane R^2 or the sphere S^2, which preserve a probability measure. For example: some commuting C^1-diffeomorphisms of S^2, which are sufficiently close to the identity and preserve a probability measure whose support is not a single point, have at least two common fixed points.

On mixing diffeomorphisms of the disk

We prove that a real analytic pseudo-rotation $f$ of the disc or the sphere is never topologically mixing. When the rotation number of $f$ is of Brjuno type, the latter follows from a KAM theorem of Rüssmann on the stability of real analytic elliptic fixed points. In the non-Brjuno case, we prove that a pseudo-rotation of class $C^k$, $k\geq 2$, is $C^{k-1}$-rigid using the simple observation, derived from Franks' Lemma on free discs, that a pseudo-rotation with small rotation number compared to its $C^1$ (or Hölder) norm must be close to Identity. From our result and a structure theorem by Fr…

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: the uniform norm of the differential of its n-th iteration and the word length of its n-th iteration. In the latter case we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds on the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology…

全部、大多数还是少数？关于区间上被扭曲和/或共轭于自身幂次的微分同胚

All, most, some? On diffeomorphisms of the interval that are distorted and/or conjugate to powers of themselves

本研究探讨区间微分同胚共轭于自身幂次的问题，分析了不同正则性下相关集合的稠密性与泛型性等性质，补充了区间微分同胚群扭曲元素的相关结果。

Distortion and Tits alternative in smooth mapping class groups

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting of diffeomorphisms which are isotopic to the identity on S does not contain any distorted elements. Moreover, we prove a weak Tits alternative for these groups.

Knot as a complete invariant of a Morse-Smale 3-diffeomorphism with four fixed points

Lens spaces are the only 3-manifolds that admit gradient-like flows with four fixed points. This is an immediate corollary of Morse inequality and of the Morse function with four critical points existence. A similar question for gradient-like diffeomorphisms is open. Solution can be approached by describing a complete topological conjugacy invariant of the class of considered diffeomorphisms and constructing of representative diffeomorphism for every conjugacy class by the abstract invariant. Ch. Bonnati and V. Z. Grines proved that the topological conjugacy class of Morse-Smale flows with uni…

关于不同维数球面乘积上$C^1$自映射的周期结构

On the periodic structure of $C^1$ self-maps on the product of spheres of different dimensions

该研究利用Lefschetz不动点理论，刻画了不同维数球面乘积上三类$C^1$自映射的周期结构，并将结果推广至紧连通单连通单Lie群上的同类映射。

On stable transitivity of finitely generated groups of volume preserving diffeomorphisms

In this paper, we provide a new criterion for the stable transitivity of volume preserving finite generated group on any compact Riemannian manifold. As one of our applications, we generalised a result of Dolgopyat and Krikorian in \cite{DK} and obtained stable transitivity for random rotations on the sphere in any dimension. As another application, we showed that for $\infty \geq r \geq 2$, any $C^r$ volume preserving partially hyperbolic diffeomorphism $g$ on any compact Riemannian manifold $M$ having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer $K$…

Calabi quasimorphism and quantum homology

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.

关于紧致2-流形上所有非游荡点均为周期点的微分同胚

On diffeomorphisms of compact 2-manifolds with all nonwandering points periodic

本文研究紧致2-流形上离散动力系统中所有非游荡点均为周期点的条件，通过构造Kupka--Smale反例并给出Hénon族的充分条件，从正反两面推进了二维情形的公开问题。

PFH spectral invariants on the two-sphere and the large scale geometry of Hofer's metric

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also obtain, as a corollary, that the group of area-preserving diffeomorphisms of the…

On primes and period growth for Hamiltonian diffeomorphisms

Here we use Vinogradov's prime distribution theorem and a multi-dimensional generalization due to Harman to strengthen some recent results concerning the periodic points of Hamiltonian diffeomorphisms. In particular we establish resonance relations for the mean indices of the fixed points of Hamiltonian diffeomorphisms which do not have periodic points with arbitrarily large periods in $\mathbb{P}^2$, the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. As an application of these results we partially recover, using only symplectic too…

Only rational homology spheres admit $Ω(f)$ to be union of DE attractors

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $Ω(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expanding maps are expanding on (co)homologies.

具有指定临界正则性的有限生成微分同胚群（在所有维度中）

Finitely generated groups of diffeomorphisms with prescribed critical regularity in every dimension

该论文构造了嵌入到指定临界正则性 $C^\alpha$ 微分同胚群但不嵌入更高正则性群的有限生成群，适用于球面和实射影空间。

The $L^p$-diameter of the group of area-preserving diffeomorphisms of $S^2$

We show that for each $p \geq 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasi-morphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_4(\mathbb{C} P^1) \to \mathcal{M}_{0,4} \cong \mathbb{C} P^1 …

Diffeomorphisms groups of tame Cantor sets and Thompson-like groups

The group of $\mathcal C^1$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations $nV$ of Thompson's group $V$ arise when we consider products of central ternary Cantor sets. We derive that the $\mathcal C^2$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

Burnside problem for groups of homeomorphisms of compact surfaces

A group $Γ$ is said to be periodic if for any $g$ in $Γ$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $\mathrm{O}(3,\R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose …

C*-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Let β: S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S^n, β) there is a Cantor minimal system (X, α) such that the corresponding product system (X x S^n, αx β) is minimal and the resulting crossed product C*-algebra C(X x S^n) \rtimes_{αx β} \mathbb{Z} is tracially approximately an interval algebra (TAI). This entails classification for such C*-algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X x S^n) \rtimes_β \m…

Distortion in transformation groups

We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(S^n), thought of as a discrete group. An appendix by Y de Cornulier shows that Homeo(S^n) has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(S^n) on a metric space by isometries has bounded orbits.

通过对称化研究球面上保面积映射的 Hofer 增长二分性

A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization

本文提出 Hamiltonian 对称化技术，证明二维球面上自治 Hamiltonian 流的 Hofer 范数要么线性增长、要么受普适常数约束。

A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms on fixed points

We define boundedness properties on the contractible fixed points set of the time-one map of an identity isotopy on a closed oriented surface with genus $g\geq1$. In symplectic geometry, a classical object is the notion of action function, defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. We give a dynamical interpretation of this function that permits us to generalize it in the case of a homeomorphism isotopic to identity that preserves a Borel finite measure of rotation vector zero, provided that a boundedness condition is satisfied. We give some p…