Area-Preserving Surface Diffeomorphisms

Volume Preserving Diffeomorphisms with Inverse Shadowing

Let f be a volume-preserving diffeomorphism of a closed C^\infty n-dimensional Riemannian manifold M: In this paper, we prove the equivalence between the following conditions: (a) f belongs to the C1-interior of the set of volume-preserving diffeoeomorphisms which satisfy the inverse shadowing property with respect to the continuous methods. (b) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the weak inverse shadowing property with respect to the continuous methods. (c) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms whic…

Entropy zero area preserving diffeomorphisms of $S^2$

In this paper we formulate and prove a structure theorem for area preserving diffeomorphisms of genus zero surfaces with zero entropy. As an application we relate the existence of faithful actions of a finite index subgroup of the mapping class group of a closed surface $Σ_g$ on $S^2$ by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups $MCG(S, \partial S)$ with non-trivial first cohomology.

Transitivity of generic semigroups of area-preserving surface diffeomorphisms

We prove that the action of the semigroup generated by a $C^r$ generic pair of area-preserving diffeomorphisms of a compact orientable surface is transitive.

Area preserving surface diffeomorphisms with polynomial decay of correlations are ubiquitous

We show that any surface admits an area preserving $C^{1+β}$ diffeomorphism with non-zero Lyapunov exponents which is Bernoulli and has polynomial decay of correlations. We establish both upper and lower polynomial bounds on correlations. In addition, we show that this diffeomorphism satisfies the Central Limit Theorem and has the Large Deviation Property. Finally, we show that the diffeomorphism we constructed possesses a unique hyperbolic Bernoulli measure of maximal entropy with respect to which it has exponential decay of correlations.

Generic area-preserving reversible diffeomorphisms

Let M be a surface and R an involution in M whose set of fixed points is a submanifold with dimension 1 and such that R is an isometry. We will show that there is a residual subset of C1 area-preserving R-reversible diffeomorphisms which are either Anosov or have zero Lyapunov exponents at almost every point.

Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms

The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.

Homoclinic orbits for area preserving diffeomorphisms

We show that C^r generically in the space of C^r conservative diffeomorphisms of a compact surface, every hyperbolic periodic point has a transverse homoclinic orbit

Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion

We construct a $C^\infty$ area-preserving diffeomorphism of the two-dimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose rotation set is $\{(0,0)\}$, which in addition has the property that almost every orbit by the lifted dynamics is unbounded and accumulates in every direction of the circle at infinity.

Generic Continuity of Metric Entropy for Volume-preserving Diffeomorphisms

Let $M$ be a compact manifold and $\text{Diff}^1_m(M)$ be the set of $C^1$ volume-preserving diffeomorphisms of $M$. We prove that there is a residual subset $\mathcal {R}\subset \text{Diff}^1_m(M)$ such that each $f\in \mathcal{R}$ is a continuity point of the map $g\to h_m(g)$ from $\text{Diff}^1_m(M)$ to $\mathbb{R}$, where $h_m(g)$ is the metric entropy of $g$ with respect to volume measure $m$.

Equality of pressures for diffeomorphisms preserving hyperbolic measures

For a diffeomorphism which preserves a hyperbolic measure the potential $ϕ^u=-\log|{\rm Jac} df|_{E^u}|$ is studied. Various types of pressure of $ϕ^u$ are introduced. It is shown that these pressures satisfy a corresponding variational principle.

A $C^{\infty}$ closing lemma on torus

Asaoka & Irie recently proved a $C^{\infty}$ closing lemma of Hamiltonian diffeomorphisms of closed surfaces. We reformulated their techniques into a more general perturbation lemma for area-preserving diffeomorphism and proved a $C^{\infty}$ closing lemma for area-preserving diffeomorphisms on a torus that is isotopic to identity. i.e., we show that the set of periodic orbits is dense for a generic diffeomorphism isotopic to identity area-preserving diffeomorphism on torus. The main tool is the flux vector of area-preserving diffeomorphisms which is, different from Hamiltonian cases, non-zero…

Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk

Given a compactly supported area-preserving diffeomorphism of the disk, we prove an integral formula relating the asymptotic action to the asymptotic winding number. As a corollary, we obtain a new proof of Fathi's integral formula for the Calabi homomorphism on the disk.

Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.

一个保持体积的、具有任意多个遍历分量且接近恒等映射的非一致双曲微分同胚

A volume preserving nonuniformly hyperbolic diffeomorphism with arbitrary number of ergodic components and close to the identity

本文证明任意维数至少为5的紧致光滑黎曼流形上存在保持体积的非一致双曲微分同胚，其遍历分量个数可任意指定（包括无穷），且可任意接近恒等映射。

Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action

The uniform norm of the differential of the n-th iteration of a diffeomorphism is called the growth sequence of the diffeomorphism. In this paper we show that there is no lower universal growth bound for volume preserving diffeomorphisms on manifolds with an effective two dimensions torus action by constructing a set of volume-preserving diffeomorphisms with arbitrarily slow growth.

Mean action of periodic orbits of area-preserving annulus diffeomorphisms

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary of the annulus, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action …

带边界紧曲面的保面积微分同胚的通用等分布性

Generic equidistribution for area-preserving diffeomorphisms of compact surfaces with boundary

本文证明了带非空边界紧曲面的通用保面积微分同胚具有等分布的周期轨道集，并给出了“通用稠密性”定理的自包含证明，将平均作用不等式扩展至任意带边界紧曲面。

A new measure of instability and topological entropy of area-preserving twist diffeomorphisms

We introduce a new measure of instability of area-preserving twist diffeomorphisms, which generalizes the notions of angle of splitting of separatrices, and flux through a gap of a Cantori. As an example of application, we establish a sharp >0 lower bound on the topological entropy in a neighbourhood of a hyperbolic, unique action-minimizing fixed point, assuming only no topological obstruction to diffusion, i.e. no homotopically non-trivial invariant circle consisting of orbits with the rotation number 0. The proof is based on a new method of precise construction of positive entropy invariant…

Measure Preserving Diffeomorphisms of the Torus are Unclassifiable

The isomorphism problem in ergodic theory was formulated by von Neumann in 1932 in his pioneering paper Zur Operatorenmethode in der klassischen Mechanik (Ann. of Math. (2), 33(3):587--642, 1932). The problem has been solved for some classes of transformations that have special properties, such as the collection of transformations with discrete spectrum or Bernoulli shifts. This paper shows that a general classification is impossible (even in concrete settings) by showing that the collection $E$ of pairs of ergodic, Lebesgue measure preserving diffeomorphisms $(S,T)$ of the 2-torus that are is…

莫比乌斯带上保密度微分同胚群的拟态射

Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band

本文首次研究非可定向流形上保密度微分同胚群，证明莫比乌斯带上的该群存在可数多个线性无关的无界拟态射，并证明紧致连通带边非可定向曲面上的保密度微分同胚群弱可缩。

Dimension approximation for diffeomorphisms preserving hyperbolic SRB measures

For a C^{1+α} diffeomorphism f preserving a hyperbolic ergodic SRB measure μ, Katok's remarkable results assert that μcan be approximated by a sequence of hyperbolic sets \{Λ_n\}_{n\geq1}. In this paper, we prove the Hausdorff dimension for Λ_n on the unstable manifold tends to the dimension of the unstable manifold. Furthermore, if the stable direction is one dimension, then the Hausdorff dimension of μcan be approximated by the Hausdorff dimension of Λ_n. To establish these results, we utilize the u-Gibbs property of the conditional measure of the equilibrium measure of -ψ^{s}(\cdot,f^n) and…

体积保持微分同胚群上的近似可控性

Approximate controllability on the group of volume-preserving diffeomorphisms

研究体积保持微分同胚群在二维和三维托罗斯上的近似可控性问题，通过分析无散向量场和常数向量场的控制作用，探讨系统的可控性条件与结果。

Differentiable conjugacy for groups of area preserving circle diffeomorphisms

We study groups of circle diffeomorphisms whose action on the cylinder $\mathcal C=\mathbb S^1\times \mathbb S^1\setminus Δ$ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of $\rm{PSL}(2,\mathbb R)$, then discuss the existence of a differentiable conjugacy.

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 $K$…

Simplicity of the group of compactly supported area preserving homeomorphisms of the open disc and fragmentation of symplectic diffeomorphisms

In 1980, Albert Fathi asked whether the group of area-preserving homeomorphisms of the 2-disc that are the identity near the boundary is a simple group. In this paper, we show that the simplicity of this group is equivalent to the following fragmentation property in the group of compactly supported, area preserving diffeomorphisms of the plane: there exists a constant m such that every element supported on a disc D is the product of at most m elements supported on topological discs whose area are half the area of D.

A complete topological classification of Morse-Smale diffeomorphisms on surfaces: a kind of kneading theory in dimension two

In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a scheme. Here we will assign to any MS surface diffeomorphism a finite amount of data which completely determines its topological conjugacy class. Moreover, we show that associated to any abstract version of this data, there exists a unique conju…

Constraints On Dynamics Preserving Certain Hyperbolic Sets

We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First if a set is a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, then it is a union of transitive, expanding attractors (or contracting repellers) of codimension 1 for any diffeomorphism such that it is hyperbolic. Secondly, if a set is a nonwandering, locally maximal, compact hyperbolic set for a surface diffeomorphism, then it is locally maximal for any diffeomorphism for which it is hyperbolic.

Polynomial growth of the derivative for diffeomorphisms on tori

We consider area--preserving diffeomorphisms on tori with zero entropy. We classify ergodic area--preserving diffeomorphisms of the 3--torus for which the sequence $\{Df^n\}_{n\in{\Bbb N}}$ has polynomial growth. Roughly speaking, the main theorem says that every ergodic area--preserving $C^2$--diffeomorphism with polynomial uniform growth of the derivative is $C^2$--conjugate to a 2--steps skew product of the form \[\tor^3\ni(x_1,x_2,x_3)\mapsto (x_1+α,\ep x_2+β(x_1),x_3+γ(x_1,x_2))\in\tor^3,\] where $\ep=\pm 1$. We also indicate why there is no 4--dimensional analogue of the above result. Ra…

Homomorphisms on groups of volume-preserving diffeomorphisms via fundamental groups

Let $M$ be a closed manifold. Polterovich constructed a linear map from the vector space of quasi-morphisms on the fundamental group $π_{1}(M)$ of $M$ to the space of quasi-morphisms on the identity component ${\rm Diff}_Ω^{\infty} (M)_{0}$ of the group of volume-preserving diffeomorphisms of $M$. In this paper, the restriction $H^{1}(π_{1}(M); {\mathbb R})\to H^{1}({\rm Diff}_Ω^{\infty} (M)_{0} ; {\mathbb R})$ of the linear map is studied and its relationship with the flux homomorphism is described.

Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk via braid groups

Recently Gambaudo and Ghys proved that there exist infinitely many quasi-morphisms on the group ${\rm Diff}_Ω^\infty (D^2, \partial D^2)$ of area-preserving diffeomorphisms of the 2-disk $D^2$. For the proof, they constructed a homomorphism from the space of quasi-morphisms on the braid group to the space of quasi-morphisms on ${\rm Diff}_Ω^\infty (D^2, \partial D^2)$. In this paper, we study the homomorphism and prove its injectivity.

Nonuniform Hyperbolicity, Global Dominated Splittings and Generic Properties of Volume-Preserving Diffeomorphisms

We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point, or the set of points with only non-zero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independe…

Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms

We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic poin…

Reducing conjugacy in the full diffeomorphism group of R to conjugacy in the subgroup of orientation-preserving maps

Let $\Diffeo=\Diffeo(\R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\R$, under the operation of composition, and let $\Diffeo^+$ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements $f,g\in \Diffeo$ are conjugate in $\Diffeo$ to associated conjugacy problems in the subgroup $\Diffeo^+$. The main result concerns the case when $f$ and $g$ have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to t…

The Lyapunov exponents of generic volume preserving and symplectic systems

We show that the integrated Lyapunov exponents of $C^1$ volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all $C^1$ symplectic diffeomorphisms on…

Weakly mixing diffeomorphisms preserving a measurable Riemannian metric are dense in $\mathcal{A}_α\left(M\right)$ for arbitrary Liouvillean number $α$

We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing $C^{\infty}$-diffeomorphisms which preserve both a smooth volume $ν$ and a measurable Riemannian metric is dense in $\mathcal{A}_α \left(M\right)= \overline{ \left\{h \circ S_α \circ h^{-1} : h \in \text{Diff}^{\infty}\left(M, ν\right) \right\}}^{C^{\infty}}$ for every Liouvillean number $α$. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly c…

保面积曲面微分同胚周期轨道的辫稳定性

Braid stability for periodic orbits of area-preserving surface diffeomorphisms

本文研究了紧曲面保面积微分同胚周期轨道的辫稳定性，证明了在非退化假设下辫的同痕类在 Hofer 度量下的小 Hamiltonian 扰动中保持稳定，并得出拓扑熵下半连续的推论，推广了 Alves-Meiwes 的结果。

Horseshoes for diffeormorphisms preserving hyperbolic measures

We give extensions of Katok's horseshoe constructions, comment on related results, and provide a self-contained proof. We consider either a $C^{1+α}$ diffeomorphism preserving a hyperbolic measure or a $C^1$ diffeomorphism preserving a hyperbolic measure whose support admits a dominated splitting.

There is no complete numerical invariant for smooth conjugacy of circle diffeomorphisms

Classical results by Poincaré and Denjoy show that two orientation-preserving $C^2$ diffeomorphisms of the circle are topologically conjugate if and only if they have the same rotation number. We show that there is no possibility of getting such a complete numerical Borel invariant for the conjugacy relation of orientation-preserving circle diffeomorphisms by homeomorphisms with higher degree of regularity. For instance, we consider conjugacy by Hölder homeomorphisms or by $C^k$-diffeomorphisms with $k\in \mathbb{Z}^+ \cup \{\infty\}$. The proof combines techniques from Descriptive Set Theory …

Fixed points of nilpotent actions on ${\mathbb S}^{2}$

We prove that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb S}^{2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb R}^{2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks, Handel and Parwani for the abelian case. We show that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\mathbb S}^{2}$ that has a finite orbit of odd cardinality also has a global fixed point. M…

Transitivity of conservative diffeomorphisms isotopic to Anosov on $\mathbb{T}^3$

We prove transitivity for volume preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^3$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.

A hyperbolic diffeomorphism with countably many ergodic components near identity

We construct a smooth hyperbolic volume preserving diffeomorphism on a four dimensional compact Riemannian manifold which has countably many ergodic components and is arbitrarily close to the identity map.

Morse-Smale surfaced diffeomorphisms with orientable heteroclinic

In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits.

Structural Stability for Fibrewise Anosov Diffeomorphisms on Principal Torus Bundles

We show a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.

Rotation Numbers for $S^2$ diffeomorphisms

These largely expository notes describe the properties of the function ${\cal R}$ which assigns a number to a $4$-tuple of distinct fixed points of an orientation preserving homeomorphism or diffeomorphism of $S^2$.

Hyperbolicity in the Volume Preserving Scenario

Hayashi has extended a result of Mañé, proving that every diffeomorphism $f$ which has a $C^1$-neighborhood $\mathcal{U}$, where all periodic points of any $g\in\mathcal{U}$ are hyperbolic, it is an Axiom A diffeomorphism. Here, we prove the analogous result in the volume preserving scenario.

Connected components of partition preserving diffeomorphisms

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$ preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq \infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\infty) = \cdots = S(f,1)$ for all such $f$ and that $S(f,1) \not= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\mathbb{R}$ quadratic forms.

Rigidity of measures on the torus: smooth stabilizers and entropy

For a nonlinear Anosov diffeomorphism of the 2-torus, we present examples of measures so that the group of $μ$-preserving diffeomorphisms is, up to zero-entropy transformations, cyclic. For families of equilibrium states $μ$, we strengthen this to show that the group of $μ$-preserving diffeomorphism is virtually cyclic.

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent. In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume pre…

Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior

In this paper we consider $C^{1+ε}$ area-preserving diffeomorphisms of the torus $f,$ either homotopic to the identity or to Dehn twists. We suppose that $f$ has a lift $\widetilde{f}$ to the plane such that its rotation set has interior and prove, among other things that if zero is an interior point of the rotation set, then there exists a hyperbolic $\widetilde{f}$-periodic point $\widetilde{Q}$$\in {\rm I}\negthinspace {\rm R^2}$ such that $W^u(\widetilde{Q})$ intersects $W^s(\widetilde{Q}+(a,b))$ for all integers $(a,b)$, which implies that $\bar{W^u(\widetilde{Q})}$ is invariant under int…

圆盘上的保面积微分同胚与正双曲轨道

Area-preserving diffeomorphisms on the disk and positive hyperbolic orbits

本文证明了开圆盘上保面积非退化微分同胚在具有至少2个内部周期点时必有无穷多个正双曲周期点，并将其应用于切透镜空间与切3维球面，细化了Hofer等人的结果。

C1-stably shadowable conservative diffeomorphisms are Anosov

In this short note we prove that if a symplectomorphism f is C1-stably shadowable, then f is Anosov. The same result is obtained for volume-preserving diffeomorphisms.

Center bunching without dynamical coherence

We answer a question of Burns and Wilkinson, showing that there are open families of volume-preserving partially hyperbolic diffeomorphisms which are accessible and center bunched and neither dynamically coherent nor Anosov. We also show in the volume-preserving setting that any diffeomorphism which is partially hyperbolic and Anosov may be isotoped to a diffeomorphism which is partially hyperbolic and not Anosov.

随机保体积微分同胚的负正则性混合

Negative regularity mixing for random volume preserving diffeomorphisms

本文研究随机保体积微分同胚的负正则性指数混合，给出通用判据并应用于随机微分同胚、SDE 及随机不可压缩 Navier-Stokes 平流扩散，证明理想标量湍流存在唯一平稳测度。

Dense properties of the space of the circle diffeomorphisms with a Liouville rotation number

Given any Liouville number $α$, it is shown that various subspaces are $C^\infty$-dense in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $α$.