V. Medvedev, E. Zhuzhoma
We prove that a structurally stable diffeomorphism of a closed (2m+1)-manifold has no codimension one non-orientable expanding attractors.
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The well known stability conjecture of Palis and Smale states that if a diffeomorphism is structurally stable then the chain recurrent set is hyperbolic. It is natural to ask if this type of results is true for an individual chain class, that is, whether or not every structurally stable chain class is hyperbolic. Regarding the notion of structural stability, there is a subtle difference between the case of a whole system and the case of an individual chain class. The later case is more delicate and contains additional difficulties. In this paper we prove a result of this type for the later cas…
We show that any codimension one hyperbolic attractor of a diffeomorphism of a (d+1)-dimensional closed manifold is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the non-orientable case, to a space with a 2 to 1 covering by such a torus-less-points. Furthermore, we show that each orientable attractor is homeomorphic to a tiling space associated to an aperiodic tiling of Rd, but that the converse is generally not true. This work allows the definition of a new invariant for aperiodic tilings, in many cases finer than the cohomological or K-theoretic…
On isolated periodic points of diffeomorphisms with expanding attractors of codimension 1
研究$Ω$-稳定3-微分同胚中孤立周期点与余维1扩张吸引子的关系，利用吸引子结构估计孤立周期点的最小数量。
In this paper, we define codimension one Anosov actions of $\RR^k, k\geq 2,$ on a closed connected orientable manifold $M$. We prove that if the ambient manifold has dimension greater than $k+2$, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension one Anosov flows.
In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the homoclinic classes is not innately locally maximal, it is hard to answer whether structurally stable homoclinic classes are hyperbolic. In this article, we make some progress on this question. We prove that if a homoclinic class is structurally stable, then it admits a dominated splitting. Moreover we prove that codimension one structurally stable classes are hyperbolic. Also, if the diffeo…
In this paper, we consider codimension one Anosov actions of IR^k, k ? 1, on closed connected orientable manifolds of dimension n+k with n? 3. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one 1-parameter subgroup of IR^k admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension n. As a byproduct, generalizing a Theorem by Ghys in the case k=1, we show that, under some assumptions abou…
On existence of expanding attractors with different dimensions
本文证明球面、环面及任意闭流形上存在具有不同维数扩张吸引子的结构稳定微分同胚或公理A系统，涵盖可定向与非可定向情形。
In this paper, we define the recurrence and "non-wandering" for decompositions. The following inclusion relations hold for codimension one foliations on closed $3$-manifolds: $\{$minimal$\} \sqcup \{$compact$\}$ $\subsetneq$ $\{$pointwise almost periodic$\}$ $\subsetneq$ $\{$recurrent$\}$ $\subsetneq$ $\{$non-wandering$\}$ $\subsetneq$ $\{$Reebless$\}$. A non-wandering codimension one $C^2$ foliation on a closed connected $3$-manifold which has no leaf with uncountably many ends is minimal (resp. compact) if and only if it has no compact (resp. locally dense) leaves. In addition, the fundament…
We study a topological structure of a closed $n$-manifold $M^n$ ($n\geq 3$) which admits a Morse-Smale diffeomorphism such that codimension one separatrices of saddles periodic points have no heteroclinic intersections different from heteroclinic points. Also we consider gradient like flow on $M^n$ such that codimension one separatices of saddle singularities have no intersection at all. We show that $M^n$ is either an $n$-sphere $S^n$, or the connected sum of a finite number of copies of $S^{n-1}\otimes S^1$ and a finite number of special manifolds $N^n_i$ admitting polar Morse-Smale systems.…
When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $\textbf{B}$, wh…
On a classification of axiom A diffeomorphisms with codimension one basic sets and isolated saddles
本文对闭可定向流形上具有余维一基本集和孤立鞍点的公理A微分同胚，基于支撑流形的拓扑结构，在非游荡集上按全局共轭进行了分类。
For the class of anisotropic Kepler problems in $\RR^d\setminus\{0\}$ with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are structurally unstable and appear only for a codimension-one submanifold of such potentials. We give them a variational characterization in terms of the behavior of the parameter-free minimizers of an associated obstacle problem. We then give a full charact…
It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom $A$ and the topology of the ambient manifold. In the given article, this statement is considered for the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable surfaces such that their non-wandering set consists of $k_f\geq 2$ connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism $f\in \mathbb G(M^2)$ is homeomorphic to the connected sum of $k_f$ closed orientable surfaces and $l_f$ two-di…
Nonsingular structural stable chaotic 3-flows of attractor-repeller type
本研究针对可定向闭3-流形，构造了非游荡集含2维扩张吸引子与有限排斥周期轨道的结构稳定非奇异流，明确了3维球面及一般闭3-流形上排斥周期轨道的纽结/链环类型约束。
Closed $4$--Manifolds Foliated by Hyperplanes
证明带有余维一叶状结构且叶为 $\mathbb{R}^3$ 的闭可定向 $4$--流形同胚于 $4$--环面 $\mathbb{T}^4$，若光滑结构允许光滑定义 $1$--形式则微分同胚。
Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and …
We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$ discrete-time hyperbolic dynamics ($r>1$), obtaining anisotropic Banach or Hilbert spaces on which the resolvent of the generator of weighted transfer operators for the flow is quasi-compact. We apply this to study the ergodic integrals of the horocycle flows $h_ρ$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov flow with orientable strong-stable distribution, we establish power-law convergence of the ergodic average. We thereby implement the program of G…
A heterodimensional cycle consists of a pair of heteroclinic connections between two saddle periodic orbits with unstable manifolds of different dimensions. Recent theoretical work on chaotic dynamics beyond the uniformly hyperbolic setting has shown that heterodimensional cycles may occur robustly in diffeomorphisms of dimension at least three. We study a concrete example of a heterodimensional cycle in the continuous-time setting, specifically in a four-dimensional vector field model of intracellular calcium dynamics. By employing advanced numerical techniques, Zhang, Krauskopf and Kirk [Dis…
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