Separatrix splitting from the point of view of symplectic geometry

Generally, the invariant Lagrangian manifolds (stable and unstable separatrices) asymptotic with respect to a hyperbolic torus of a Hamiltonian system do not coincide. This phenomenon is called separatrix splitting. In this paper, a symplectic invariant qualitatively describing separatrix splitting for hyperbolic tori of maximum (smaller by one than the number of degrees of freedom) dimension is constructed. The construction resembles that of the homoclinic invariant found by lazutkin for two-dimensional symplectic maps and of Bolotin's invariant for splitting of asymptotic manifolds of a fixed point of a symplectic diffeomorphism. Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 890–906, June, 1997. Translated by O. V. Sipacheva     

Stably average shadowable homoclinic classes

Let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold M, and let H_f(p) be the homoclinic class of f containing p. In this paper, we show that if H_f(p) is locally maximal and every hyperbolic periodic point in H_f(p) is uniformly far away from being nonhyperbolic, and H_f(p) has the average shadowing property, then H_f(p) is hyperbolic.     

Separating diffeomorphisms of the torus

There exists a diffeomorphism on the n-dimensional torus Tⁿ which is conjugate with a hyperbolic linear automorphism, but is not an Anosov diffeomorphism. A diffeomorphismf: Tⁿ→Tⁿ has such a property iff is separating and belongs to the C⁰ closure of the Anosov diffeomorphisms.     

Local product structure for expansive homeomorphisms

Let be an expansive homeomorphism with dense topologically hyperbolic periodic points, M a closed manifold. We prove that there is a local product structure in an open and dense subset of M. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus.     

On the simple isotopy class of a source-sink diffeomorphism on the 3-sphere

The results obtained in this paper are related to the Palis-Pugh problem on the existence of an arc with finitely or countably many bifurcations which joins two Morse-Smale systems on a closed smooth manifold M ⁿ . Newhouse and Peixoto showed that such an arc joining flows exists for any n and, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. For n = 1, this is related to the presence of the Poincaré rotation number, and for n = 2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimension n = 3, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse-Smale diffeomorphism on the 3-sphere without heteroclinic intersections to be joined by a simple arc with a “source-sink” diffeomorphism are also found.     

Quasi-energy function for diffeomorphisms with wild separatrices

We consider the class G ₄ of Morse—Smale diffeomorphisms on $ \mathbb{S} $ ³ with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse—Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class G _4,1 ? G ₄ of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere $ \mathbb{S} $ ³. For each diffeomorphism in G _4,1, we present a quasi-energy function with six critical points.     

Weakly Hyperbolic Actions of Kazhdan Groups on Tori

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C² volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we restrict our attention to weakly hyperbolic actions on the torus. When the acting group is a lattice subgroup of a semisimple Lie group with no compact factors and all (almost) simple factors of real rank at least two, we show that weak hyperbolicity in the original action implies weak hyperbolicity for the induced action on the fundamental group. As a corollary, we obtain that any such action on the torus is continuously semiconjugate to the affine action coming from the fundamental group via a map unique in the homotopy class of the identity. Under the additional assumption that some partially hyperbolic group element has quasi-isometrically embedded lifts of unstable leaves to the universal cover, we obtain a conjugacy, resulting in a continuous classification for these actions. Partially funded by VIGRE grant DMS-9977371 Received: January 2005 Revision: August 2005 Accepted: September 2005     

Oscillations near a separatrix in the duffing equation

A small periodic perturbation results in a complicated dynamics near separatrices and saddle points. A two-parameter family of asymptotic solutions staying near separatrices for a long time is constructed. Solutions from this family depend nonsmoothly on the perturbation parameter. An example is given in which the values of the perturbation parameter for this family of solutions are determined by a set with structure of the type of the Cantor set.     

Hyperbolic annulus principle

We study a special class of diffeomorphisms of an annulus (the direct product of a ball in ? ^k , k ≥ 2, by an m-dimensional torus). We prove the so-called annulus principle; i.e., we suggest a set of sufficient conditions under which each diffeomorphism in a given class has an m-dimensional expanding hyperbolic attractor.     

Diffeomorphisms preserving R‐circles in three dimensional CR manifolds

R ‐circles in (non‐degenerate) three dimensional CR manifolds are the analogues to traces of Lagrangian totally geodesic planes on S³ viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain Legendrian curves on the manifold. We prove that a diffeomorphism between three dimensional CR manifolds which preserve circles is either a CR diffeomorphism or a conjugate CR diffeomorphism.     

The splitting of separatrices,the branching of solutions and non-integrability in the problem of the motion of a spherical pendulum with an oscillating suspension point

The effectiveness of the results obtained previously in [Dovbysh SA. Transversal intersection of separatrices and non-existence of an analytical integral in multidimensional systems. In: Ambrosetti A, Dell Antonio GF, editors. Variational and Local Methods in the Study of Hamiltonian Systems. Singapore, etc: World Scientific; 1995. p. 156–65; Dovbysh SA. Transversal intersection of separatrices, the structure of a set of quasi-random motions and the non-existence of an analytic integral in multidimensional systems. Uspekhi Mat Nauk 1996; 51(4): 153–54; Dovbysh SA. Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: Separatrices of hyperbolic periodic points. Collect Math 1999; 50(2): 119–97; Dovbysh SA. Branching of the solutions in the complex domain from the point of view of symbolic dynamics and the non-integrability of multidimensional systems. Dokl Ross Akad Nauk 1998; 361(3): 303–6] on the non-integrability of multidimensional systems is illustrated using the example of the problem of the motion of a spherical pendulum with a suspension point performing small periodic oscillations. With this aim, the splitting of the separatrices of the unstable equilibrium position and the branching of the solutions are investigated. It is shown that the separatrices are split for any law of motion of the suspension point, and a simple criterion of the presence of their transversal intersection is obtained. The validity of the non-integrability result, based on a combination of the conditions related to the splitting of multidimensional separatrices and to the branching of the solutions, is also pointed out.     

Robust Weak Ergodicity And Stable Ergodicity

In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r(r > 1) conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive (or negative) central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r(r > 1) conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.     

Hyperbolicity, transversality and analytic first integrals

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.     

Almost commensurability of 3-dimensional Anosov flows

Two flows are almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwise almost commensurable.     

Spinning gas clouds: Liouville integrable cases

We consider the class of ellipsoidal gas clouds expanding into a vacuum [1, 2] which has been shown to be a Liouville integrable Hamiltonian system [3]. This system presents several interesting features, such as the Painlevé property [4, 5], the existence of Bäcklund transformations and the separability of variables, all shown to be present in at least several sub-cases. A remarkable result that emerged from the study of the cases of rotation around a fixed principal axis, was that the Liouville torus, which is the locus of trajectories of the representative point of the cloud when all the constants of motion are fixed, could be assimilated with a quartic surface presenting 16 conic point singularities. The geometry of such surfaces is entirely determined by the datum of a 6th degree polynomial in one variable, and the consideration of the corresponding natural coordinate system then led to the separation of variables for these cases [6]. Further, the equation of the surface takes the form of a 4×4 determinant, which constitutes a generalization of Stieltjes 4 × 4 determinant formulation of the addition formula for elliptic functions; and the corresponding matrix also defines the system of the equations of motion; so that it can be said that the differential system is completely determined by the surface’s geometry. Forsaking now the assumption of a fixed rotation axis, in cases where the energy constant takes its minimum value compatible with the other constants of motion, we found that the Liouville torus was still reducible to the form of a quartic surface, presenting 15 conic points only instead of 16 (16 conic points were indeed present originally, but one of them had to disappear in the process of reducing the surface to the 4th degree). The geometry of these surfaces is entirely determined by the datum of a plane unicursal quartic (which is the transformed version of the missing conic point). The system can be reduced to the form of a differential equation of second degree, the coefficients of which are polynomials of degree 7, which are determined by the surface’s geometry, except for their quadratic dependence on a single free parameter, z. Defining u the (time-like) independent variable, and Φ the integration constant (which are functions defined on the Liouville torus), it is found that Φ depends linearly on the parameter: Φ = Φ(z) and then u may be taken to coincide with Φ(z′), for any value of z′ distinct from z. Solving the system for one particular value of z therefore also solves it for all other values of the parameter. It appears that the geometry alone does not specify in this case any particular value of z, but then any two values lead to differential systems which (although their solutions differ) turn out to be equivalent. It may also be worth pointing out that changing z may be viewed as exchanging the roles of u and Φ. Finally, in degenerate cases the Liouville torus presents a double line of self-intersection, and the separation of variables can be achieved. Sections by planes through the double line are conic sections, which may be labeled by a parameter w, say. Denoting α the eccentric anomaly on the conic, the differential system in fact takes a remarkably simple form: da/dw = f(w), and involves an elliptic integral.     

Lower bounds for Kazhdan-Lusztig polynomials from patterns

Kazhdan-Lusztig polynomials P_x,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values P_x,w(1) in terms of "patterns". A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group G. Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties.     

Estimates for solutions of reduced hyperbolic equations of the second order with a large parameter

We consider solutions of inhomogeneous, reduced hyperbolic equations of the second order, with a large parameter multiplying the unknown function. These solutions are defined on the m-dimensional region outside a star-shaped body. They satisfy an “outgoing” radiation condition at infinity and a Dirichlet boundary condition.We obtain a priori estimates for these solutions, at every point outside or on the surface of a two- or three-dimensional star-shaped body, that hold for sufficiently large values of the parameter. We prove that each solution is bounded by a linear combination of (i) the maximum norm of its prescribed boundary values, (ii) the L₂ norm of the prescribed values of its tangential derivative, (iii) an L₂ norm of the source term. This result is based on similar inequalities that we first obtain for a certain L₂ norm of the gradient, and of the normal derivative on the boundary, of each solution defined outside an m-dimensional star-shaped body.For the special case of the reduced wave equation, Morawetz and Ludwig [1] have obtained similar estimates. Just as their results have been used in [3] to confirm the geometrical theory of diffraction, the estimates obtained in this paper can be used to establish the validity of certain formal asymptotic solutions of reduced hyperbolic equations.     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

Kupka-Smale theorem for polynomial automorphisms of ?2 and persistence of heteroclinic intersections

A map is Kupka-Smale if all periodic points are hyperbolic and the stable and unstable manifolds of any two saddle points are transverse. Here we prove that Kupka-Smale maps form a residual set of full Lebesgue measure in the space of polynomial automorphisms of ² of fixed dynamical degree d2. We also prove that a heteroclinic point of two saddle periodic orbits may be continued over (almost) the entire parameter space for this set of maps. This is one of the first persistence theorems proved in holomorphic dynamics in several variables.     

On Trajectories Entirely Situated Near a Hyperbolic Set

Let I ₁ be a set of points such that their trajectories under a diffeomorphism f ₁ are entirely close enough to a hyperbolic set F ₁ of this diffeomorphism. Then it is proved that the structure of I ₁ and restriction \( {f}_1\left|{}_{I_1}\right. \) (“motion in I ₁”) are essentially defined (up to an equivariant homeomorphism) by “internal dynamics” in F ₁ , i.e., by the restriction \( {f}_1\left|{}_{{}_{F_1}}\right. \) . (In more detail: the equivariant homeomorphism g ₁ of the set F ₁ on the hyperbolic set F ₂ of the second diffeomorphism f ₂ (probably, acting on another manifold M ₂) is extendable to an equivariant homeomorphic embedding I ₁ → M ₂ . The image of the imbedding contains all the trajectories f ₂ close enough to F ₂ .)