Reversibility in the diffeomorphism group of the real line

Locally discrete groups of analytic diffeomorphisms of the circle

We show that a finitely generated group of analytic diffeomorphisms that is expanding and locally discrete in the analytic category is analytically conjugate to a uniform lattice of a finite covering of the group of projective maps of the projective line, acting naturally on the corresponding finite covering of the projective line.

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…

On the centralizer of diffeomorphisms of the half-line

Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z^r its centralizer in the group of C^r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^1 is a one-parameter group. On the other hand, Sergeraert constructed an f whose centralizer Z^r, $2\le r\le \infty$, reduces to the group generated by f. We show that Z^r can actually be a proper dense and uncountable subgroup of Z^1 and that this phenomenon is not scarce.

Arithmetic properties of centralizers of diffeomorphisms of the half-line

Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z^r_f its centralizer in the group of C^r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^1_f is always a one-parameter group, naturally identified to \R, (with f identified to 1). On the other hand, Z^r_f, for r greater or equal to 2, can be smaller: in [Se], Sergeraert constructed an f whose C^infty centralizer reduces to the infinite cyclic group generated by f (i.e Z^\infty_f identifies to \Z). In [Ey1], we adapted Sergeraert's construction to obtain an f whose C^r centralizer, for all r…

Commutator length of annulus diffeomorphisms

We study the group of C^{r}-diffeomorphisms of the closed annulus that are isotopic to the identity. We show that, for r different from 3, the linear space of homogeneous quasi-morphisms on this group is one dimensional. Therefore, the commutator length on this group is (stably) unbounded. In particular, this provides an example of a manifold whose diffeomorphisms group is unbounded in the sense of Burago, Ivanov and Polterovich.

The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial

In the late nineties, Smale proposed a list of problems for the next century and, among these, it was conjectured that for every $r\ge 1$ a $C^r$-generic diffeomorphism has trivial centralizer. Our contribution here is to prove the triviality of $C^r$-centralizers on hyperbolic basic sets. In particular, $C^r$-generic transitive Anosov diffeomorphisms have a trivial $C^1$-centralizer. These results follow from a more general criterium for expansive homeomorphisms with the gluing orbit property. We also construct a linear Anosov diffeomorphism on $\mathbb T^3$ with discrete, non-trivial central…

Groups of germs of analytic diffeomorphisms in (C^2,0)

Let G a group of germs of analytic diffeomorphisms in (C^2,0). We find some remarkable properties supposing that G is finite, linearizable, abelian nilpotent and solvable. In particular, if the group is abelian and has a generic dicritic diffeomorphisms, then the group is a subgroup of a 1-parametric group. In addition, we study the topological behaviour of the orbits of a dicritic diffeomorphisms. Last, we find some invariants in order to know when two diffeomorphisms are formally conjugates.

Generic non-trivial resonances for Anosov diffeomorphisms

We study real analytic perturbations of hyperbolic linear automorphisms on the 2-torus. The Koopman and the transfer operator are nuclear of order 0 when acting on a suitable Hilbert space. We show the generic existence of non-trivial Ruelle resonances for both operators. We prove that some of the perturbations preserve the volume and some of them do not.

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.

Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms

We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Holder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a contionuous conjugacy between two cocycles.

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。

Moduli of 3-dimensional diffeomorphisms with saddle-foci

We consider a space $\mathcal{U}$ of 3-dimensional diffeomorphisms $f$ with hyperbolic fixed points $p$ the stable and unstable manifolds of which have quadratic tangencies and satisfying some open conditions and such that $Df(p)$ has non-real expanding eigenvalues and a real contracting eigenvalue. The aim of this paper is to study moduli of diffeomorphisms in $\mathcal{U}$. We show that, for a generic element $f$ of $\mathcal{U}$, all the eigenvalues of $Df(p)$ are moduli and the restriction of a conjugacy homeomorphism to a local unstable manifold is a uniquely determined linear conformal m…

Anosov diffeomorphisms 在 $\mathbb{T}^{3}$ 上的局部光滑刚性

Local smooth rigidity of Anosov diffeomorphisms in $\mathbb{T}^{3}$

研究在 $\mathbb{T}^{3}$ 上的 Anosov diffeomorphisms 之间 $C^0 $ 共轭时，若周期数据匹配则共轭必为光滑的性质。

On the dynamic of holomorphic diffeomorphisms groups fixing a commune point, on C^n

In this paper, we study the action on C^n of any group G of holomorphic diffeomorphisms (automorphisms) of C^n fixing 0. Suppose that there is x in C^n, having an orbit which generates C^n and also E(x)=C^n, where E(x) is the vector space generated by L_{G}={D_{0}fx, f in G }. We give an important condition so that an orbit G(x) is isomorphic (by linear map) to the orbit L_{G}(x)of the linear group L_{G}. More if G is abelian, we prove the existence of a G-invariant open set U, dense in C^n, in which every orbit O is relatively minimal (i.e. the closure of O in U is a closed non empty, G-invar…

One-parameter families of circle diffeomorphisms with strictly monotone rotation number

We show that if $f \colon S^1 \times S^1 \to S^1 \times S^1$ is $C^2$, with $f(x, t) = (f_t(x), t)$, and the rotation number of $f_t$ is equal to $t$ for all $t \in S^1$, then $f$ is topologically conjugate to the linear Dehn twist of the torus $(1&1 0&1)$. We prove a differentiability result where the assumption that the rotation number of $f_t$ is $t$ is weakened to say that the rotation number is strictly monotone in $t$.

Homomorphisms between diffeomorphism groups

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1- manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if M is any closed manifold, and Diff(M)_0 injects into the diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the answ…

Family of intersecting totally real manifolds of $(C^n ,0)$ and germs of holomorphic diffeomorphisms

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group can be linearized holomorphically near the fixed point. It rests on a "small divisors condition" of the family of linear parts. The second part of this article is devoted to the study families of totally real intersecting n-submanifolds of (C n , 0). We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it…