Stefan Müller
We prove that a compactly supported homeomorphism of a smooth manifold of dimension greater or equal to 5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given homeomorphism is in addition volume preserving, then it can be approximated uniformly by volume preserving diffeomorphisms.
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We introduce Smale A-homeomorphisms that includes regular, semi-chaotic, chaotic, and super chaotic homeomorphisms of topo\-lo\-gi\-cal $n$-manifold $M^n$, $n\geq 2$. Smale A-homeo\-mor\-p\-hisms contain A-diffeomorphisms (in particular, structurally stable diffeomorphisms) provided $M^n$ admits a smooth structure. Regular A-homeomorphisms contain all Morse-Smale diffeomorphisms, while semi-chaotic and chaotic A-homeomorphisms contain A-diffeo\-mor\-p\-hisms with trivial and nontrivial basic sets. Super chaotic A-homeo\-mor\-p\-hisms contain A-diffeomorphisms whose basic sets are nontrivial. W…
Let f be a class P -homeomorphism of the circle. We prove that there exists a piecewise analytic homeomorphism that conjugate f to a one-class P with prescribed break points lying on pairwise distinct orbits. As a consequence, we give a sharp estimate for the smoothness of a conjugation of class P -homeomorphism f of the circle satisfying the (D)-property (i.e. the product of f-jumps in the break points contained in a same orbit is trivial), to diffeomorphism. When f does not satisfy the (D)-property the conjugating homeomorphism is never piecewise C^1 and even more it is not absolutely contin…
The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $\bullet$ the generating function of such a foliation is C1 ; $\bullet$ the foliation is H{ö}lder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol'd-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugate…
Given closed topological $n$-manifold $M^n$, $n\geq 2$, one introduces the classes of Smale regular $SRH(M^n)$ and Smale semi-regular $SsRH(M^n)$ homeomorphisms of $M^n$ with $SRH(M^n)\subset~SsRH(M^n)$. The class $SRH(M^n)$ contains all Morse-Smale diffeomorphisms, while $SsRH(M^n)$ contains A-diffeomorphisms with trivial and some nontrivial basic sets provided $M^n$ admits a smooth structure. We select invariant sets that determine dynamics of Smale homeomorphisms. This allows us to get necessary and sufficient conditions of conjugacy for $SRH(M^n)$ and $SsRH(M^n)$. We deduce applications fo…
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.
We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with nonempty boundary, then the quotient of A_0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.
A partially hyperbolic diffeomorphism $f$ is structurally quasi-stable if for any diffeomorphism $g$ $C^1$-close to $f$, there is a homeomorphism $π$ of $M$ such that $π\circ g$ and $f\circπ$ differ only by a motion $τ$ along center directions. $f$ is topologically quasi-stable if for any homeomorphism $g$ $C^0$-close to $f$, the above holds for a continuous map $π$ instead of a homeomorphism. We show that any partially hyperbolic diffeomorphism $f$ is topologically quasi-stable, and if $f$ has $C^1$ center foliation $W^c_f$, then $f$ is structurally quasi-stable. As applications we obtain con…
Thurston obtained a classification of individual surface homeomorphisms via the dynamics of the corresponding mapping class elements on Teichmüller space. In this paper we present certain extended versions of this, first, to random products of homeomorphisms and second, to holomorphic self-maps of Teichmüller spaces.
A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+α}$ diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nodes) by a semi-conjugacy whose fibers carry zero entropy.
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 …
We give general classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys' "extension problem" for diffeomorphisms of manifolds with boundary, as well as a classification of all homomorphisma $\mathrm{Homeo}_0(M) \to \mathrm{Homeo}_0(N)$ when dim(M) = dim(N) (and related results for diffeomorphisms), and a complete classification of actions of $\mathrm{Homeo}_0(S^1)$ on surfaces. This resolves many problems in a program initiated b…
$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval
研究平坦段边界奇点次数不同的圆周映射类，证明其拓扑共轭若为双Lipschitz同胚则必为$C^1$微分同胚，从而两类等价。
This self-contained paper is part of a series \cite{FF2,FF3} on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: 1) Any homomorphism of (almost any) mapping class group or automorphism group of a free group into $\Diff_+^r(S^1), r\geq 2$ is trivial. For r=0 Nielsen showed that in many cases nontrivial (even faithful) representations exist. Somewhat weaker results are proven for finite index subgroups. 2) We construct a finitely-presented group of real-analytic diffeomorphisms of $\R$ which is not residually finite.
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…
On sufficient conditions for the transitivity of homeomorphisms
本文推导了具有跟踪性质的同胚拓扑传递的充要条件（含稠密不变子集的重心性质），对比Anosov微分同胚的相关条件，描述$C^1$内部、讨论例子并给出应用，为动力学传递性研究提供新依据。
In this note we prove that a homeomorphism is countably-expansive if and only if it is measure-expansive. This result is applied for showing that the $C^1$-interior of the sets of expansive, measure-expansive and continuum-wise expansive $C^1$-diffeomorphisms coincide.
We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit $n$-dimensional cube onto itself, whose Jacobian is equal to $-1$ a.e. Moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.
This note illustrates the strategy of our paper on piecewise affine surface homeomorphisms by giving a new proof of the finite multiplicity of the maximum entropy measure of Anosov diffeomorphisms (here on surfaces). This approach avoids the explicit construction of Markov partitions and will be applied elsewhere to some non-uniformly hyperbolic diffeomorphisms.
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 …
In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed po…
Let p be a saddle fixed point for an orientation-preserving surface diffeomorphism f admitting a homoclinic point q. Let V be an open 2-cell bounded by a simple loop formed by two arcs joining p to q lying respectively in the stable and unstable curves at p. It is shown that f|V has fixed point index 1 or 2 depending only on the geometry of V near p.
We examine uniqueness of equilibrium states for the natural extension of a topologically exact, non-uniformly expanding, local homeomorphism with a Hölder continuous potential function. We do this by applying general techniques developed by Climenhaga and Thompson, and show there is a natural condition on decompositions that guarantees that a unique equilibrium state exists. We then show how to apply these results to partially hyperbolic attractors.
We study the problem of existence of a periodic point in the boundary of an invariant domain for a surface homeomorphism. In the area-preserving setting, a complete classification is given in terms of rationality of Carathéordory's prime ends rotation number, similar to Poincaré's theory for circle homeomorphisms. In particular, we prove the converse of a classic result of Cartwright and Littlewood. This has a number of consequences for generic area preserving surface diffeomorphisms. For instance, we extend previous results of J. Mather on the boundary of invariant open sets for $C^r$-generic…
Regularity, linear response formula and differentiability of the free energy for non-uniformly expanding local homeomorphisms
本文研究了一类非一致扩张局部同胚的平衡态，证明了平衡态的存在唯一性及统计量关于动力系统的可微性。
We show that, in many situations, a homeomorphism $f$ of a manifold $M$ may be recovered from the (marked) isomorphism class of a finitely generated group of homeomorphisms containing $f$. As an application, we relate the notions of {\em critical regularity} and of {\em differentiable rigidity}, give examples of groups of diffeomorphisms of 1-manifolds with strong differential rigidity, and in so doing give an independent, short proof of a recent result of Kim and Koberda that there exist finitely generated groups of $C^α$ diffeomorphisms of a 1-manifold $M$, not embeddable into $\mathrm{Diff}…
Using an approach due to Bowen, Franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the Bowen property. We show that this conclusion remains true using weaker non-uniform versions of specification, expansivity, and the Bowen property. We also establish a corresponding result for homeomorphisms. In the homeomorphism case, we obtain the upper bound from the level-2 large deviations principle for the unique equilibrium state. The theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic for…
We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $δ> 0$ such that every $δ$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient d…
Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_0$ with Lefschetz index one. Le Roux conjectured that $z_0$ is accumulated by periodic orbits. In this article, we will approach Le Roux's conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_0$ and if the area of $M$ is finite, then $z_0$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_0$ is the limit in weak-star topology of a sequence of invaria…
We use the Invariance Principle of Avila and Viana to prove that every partially hyperbolic symplectic diffeomorphism with 2-dimensional center bundle, and satisfying certain pinching and bunching conditions, can be $C^r$-approximated by non-uniformly hyperbolic diffeomorphisms.
In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those with support in balls and conjugation-generated norm.
On uniformly quasiconformal Anosov diffeomorphisms with two dimensional distributions
本研究证明具有二维不稳定分布的传递一致$u$-拟共形Anosov微分同胚有全局稳定和乐，由此去除Kalinin-Sadovskaya定理的额外假设，推导出该类微分同胚与下环面仿射Anosov微分同胚$C^\infty$共轭。
Ghys and Sergiescu proved in the $80$s that Thompson's group $T$, and hence $F$, admits actions by $C^{\infty}$ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of $C^\infty$ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the group…
The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal identity isotopy such that we can give a fine descriptions of the dynamics of its transverse foliation. We will define a kind of identity isotopies: torsion-low isotopies. In particular, when $f$ is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy $I$ of $f$ is torsion-low if and only if for every point $z$ fixed along the isotopy, the (real) rotation number $ρ(I,z)$, which is well defin…
We prove that the system resulting of coupling the standard map with a fast hyperbolic system is robustly non-uniformly hyperbolic.
We study the $C^1$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^2$ partially hyperbolic symplectic systems which have bounded $C^2$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.
In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the $C^1$ topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.
In this paper we will show that if a sequence of natural numbers satisfies a certain growth rate, then there is a weak mixing diffeomorphism on $\mathbb{T}^2$ that is uniformly rigid with respect to that sequence. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly defined conjugation maps and the constructions are done in the $C^{\infty}$-topology as well as in the real-analytic topology.
We give a complete description of the behaviour of the sequence of displacements $η_n(z)=Φ^n(x) - Φ^{n-1}(x) \ \rmod \ 1$, $z=\exp(2π\rmi x)$, along a trajectory $\{φ^{n}(z)\}$, where $φ$ is an orientation preserving circle homeomorphism and $Φ:\mathbb{R} \to \mathbb{R}$ its lift. If the rotation number $\varrho(φ)=\frac{p}{q}$ is rational then $η_n(z)$ is asymptotically periodic with semi-period $q$. This convergence to a periodic sequence is uniform in $z$ if we admit that some points are iterated backward instead of taking only forward iterations for all $z$. If $\varrho(φ) \notin \mathbb{Q…
We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of O. Sarig for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.
We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a certain degree of freedom for the choice of the second one. For a non-countable set of choices, this leaf is not periodic and the diffeomorphism is Kupka-Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. As a consequence of our construction,…
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。
Beyond topological hyperbolicity: the L-shadowing property
本文研究紧空间动力系统的L-阴影性质，证明结构稳定微分同胚和部分伪Anosov微分同胚满足该性质，并借助谱分解、拓扑半马蹄及正有限可扩性给出其刻画与推论。
We prove that if an area-preserving homeomorphism of the torus in the homotopy class of the identity has a rotation set which is a nondegenerate vertical segment containing the origin, then there exists an essential invariant annulus. In particular, some lift to the universal covering has uniformly bounded displacement in the horizontal direction.
We consider $C^{1+ε}$ diffeomorphisms of the torus, denoted $f,$ homotopic to the identity and whose rotation sets have interior. We give some uniform bounds on the displacement of points in the plane under iterates of a lift of $f,$ relative to vectors in the boundary of the rotation set and we use these estimates in order to prove that if such a diffeomorphism $f$ preserves area, then the rotation vector of the area measure is an interior point of the rotation set. This settles a strong version of a conjecture proposed by P. Boyland. We also present some new results on the realization of ext…
We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple…
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