Eric Bedford, John Smillie
This paper continues our investigation of the dynamics of polynomial diffeomorphisms of C^2. We introduce a dynamical property of polynomial diffeomorphisms that generalizes hyperbolicity in the way that semi-hyperbolicity generalizes hyperbolicity for polynomial maps of the complex plane.
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Quasi-Anosov diffeomorphism is a kind of important dynamical system due to R. Mane 1970s. In this paper, we present a criterion for such dynamics using ergodic theory.
Let $M$ be an $n$-dimensional manifold supporting a quasi Anosov diffeomorphism. If $n=3$ then either $M={\mathbb T}^3$, in which case the diffeomorphisms is Anosov, or else its fundamental group contains a copy of ${\mathbb Z} ^6$. If $n=4$ then $Π_1(M)$ contains a copy of ${\mathbb Z} ^4$, provided that the diffeomorphism is not Anosov.
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…
We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.
In 1969, Hirsch posed the following problem: given a diffeomorphism, and a compact invariant hyperbolic set, describe its topology and restricted dynamics. We solve the problem where the hyperbolic invariant set is a closed 3-manifold: if the manifold is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions. The dynamics of the diffeomorphisms restricted to these manifolds, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions…
A partially hyperbolic diffeomorphism $f$ has quasi-shadowing property if for any pseudo orbit ${x_k}_{k\in \mathbb{Z}}$, there is a sequence of points ${y_k}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_k)$ by a motion $τ$ along the center direction. We show that any partially hyperbolic diffeomorphism has quasi-shadowing property, and if $f$ has $C^1$ center foliation then we can require $τ$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^0$-perturbation. When…
Let $f$ be a partially hyperbolic diffeomorphism. $f$ is called has the quasi-shadowing property if for any pseudo orbit $\{x_k\}_{k\in \mathbb{Z}}$, there is a sequence $\{y_k\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ lies in the local center leaf of $f(y_k)$ for any $k\in \mathbb{Z}$. $f$ is called topologically quasi-stable if for any homeomorphism $g$ $C^0$-close to $f$, there exist a continuous map $π$ and a motion $τ$ along the center foliation such that $π\circ g=τ\circ f\circπ$. In this paper we prove that if $f$ is dynamically coherent then it has quasi-shadowing and topologi…
We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor.
In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a $C^1$-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow.
A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots
本文为广义Mazur纽结定义的Pixton微分同胚构造了具有最少临界点数的拟能量函数，并给出了Lyapunov函数临界点数量的下界估计。
Quasi-isometric center action in dimension 3
研究了3维中保持中心叶理且拟等距作用的传递部分双曲微分同胚，证明其在有限提升和迭代下仅为斜积或离散化Anosov流。
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.
Partially Hyperbolic Dynamics with Quasi-isometric Center
本文研究了闭3-流形上具有拟等距中心的偏双曲微分同胚，证明了在无su-环面时的可达性与遍历性，确认了Hertz-Hertz-Ures遍历性猜想，并给出了此类微分同胚的完整分类。
We construct an embedding $Φ$ of $[0,1]^{\infty}$ into $Ham(M, ω)$, the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold $(M, ω)$. We then prove that $Φ$ is in fact a quasi-isometry. After imposing further assumptions on $(M, ω)$, we adapt our methods to construct a similar embedding of $\mathbb{R} \oplus [0,1]^{\infty}$ into either $Ham(M, ω)$ or $\widetilde{Ham}(M, ω)$, the universal cover of $Ham(M, ω)$. Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on a continuity result…
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.
We study quasi-morphisms on the groups Pn of pure braids on n strings and on the group D of compactly supported area-preserving diffeomorphisms of an open two-dimensional disc. We show that it is possible to build quasi-morphisms on Pn by using knot invariants which satisfy some special properties. In particular, we study quasi-morphisms which come from knot Floer homology and Khovanov-type homology. We then discuss possible variations of the Gambaudo-Ghys construction, using the above quasi-morphisms on Pn to build quasi-morphisms on the group D of diffeomorphisms of a 2-disc.
Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of recently established results for non-discrete groups acting on the circle. These results are also applied to show that, for many groups as above, every continuous orbit equivalence must coincide almost everywhere with a diffeomorphism of the corresponding manifold.
Stable local dynamics: expansion, quasi-conformality and ergodicity
本文引入拟共形混合器这一几何方法，克服了稳定遍历作用仅存在于一维流形的限制，证明了闭流形上$C^{1+α}$类微分同胚的有限生成群作用及某些代数作用的稳定遍历性。
We present new examples of generic diffeomorphisms without attractors. Also, we study how these wild classes are accumulated by infinitely many other classes (obtaining that the chain recurrence classes different from the only quasi-attractor are contained in center stable manifolds). The construction relies on some derived from Anosov (DA) constructions and uses strongly the semiconjugacy obtained by these diffeomorphisms. An interesting feature of this examples is that we can show that robustly, they present a unique attractor in the sense of Milnor.
We study spatial discretizations of dynamical systems: is it possible to recover some dynamical features of a system from numerical simulations? Here, we tackle this issue for the simplest algorithm possible: we compute long segments of orbits with a fixed number of digits. We show that for every $r\ge 1$, the dynamics of the discretizations of a $C^r$ generic conservative diffeomorphism of the torus is very different from that observed in the $C^0$ regularity. The proof of our results involves in particular a local-global formula for discretizations, as well as a study of the corresponding li…
Let $f$ be a germ of holomorphic diffeomorphism of $\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is locally holomorphically linearizable if and only if there exists a particular $f$-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.
What kind of dynamics do we observe in general on the circle? It depends somehow on the interpretation of "in general". Everything is quite well understood in the topological (Baire) setting, but what about the probabilistic sense? The main problem is that on an infinite dimensional group there is no analogue of the Lebesgue measure, in a strict sense. There are however some analogues, quite natural and easy to define: the Malliavin-Shavgulidze measures provide an example and constitute the main character of this text. The first results show that there is no actual disagreement of general feat…
In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.
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