Symplectic Diffeomorphisms Research Articles

Homoclinic points of symplectic partially hyperbolic systems with 2D centre

We consider a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with area-preserving diffeomorphisms and prove that every hyperbolic periodic point has transverse homoclinic points.

Joint integrability and spectral rigidity for Anosov diffeomorphisms

AbstractLet be an Anosov diffeomorphism whose linearization is irreducible. Assume that is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable subbundle and the unstable subbundle are jointly integrable, then is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization . Moreover, admits the finest dominated splitting in the weak stable subbundle with dimensions matching those for , and it has spectral rigidity along all these subbundles. In dimension 4, we also obtain a similar result by grouping the weak stable and unstable subbundles together as a center subbundle and assuming joint integrability of the strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism that is ‐close to an irreducible nonconformal automorphism , the extremal subbundles of are jointly integrable if and only if is smoothly conjugate to .

Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface

We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (integrable elliptic islands) among analytic, symplectic diffeomorphisms in many isotopy classes of any closed surface. In particular this solves a problem introduced by F. Przytycki (1982).

On symplectic invariants of planar 3-webs

The classical web geometry [1,3,4] studies invariants of foliation families with respect to pseudogroup of diffeomorphisms. Thus for the case of planar 3-webs the basic semi invariant is the Blaschke curvature, [2]. It is also curvature of the Chern connection [4] that are naturally associated with a planar 3-web.
 In this paper we investigate invariants of planar 3-webs with respect to group of symplectic diffeomorphisms. We found the basic symplectic invariants of planar 3-webs that allow us to solve the symplectic equivalence problem for planar 3-webs in general position. The Lie-Tresse theorem, [4], gives the complete description of the field of rational symplectic differential invariants of planar 3-webs. We also give normal forms for homogeneous 3-webs, i.e. 3-webs having constant basic invariants.

On the divergence of Birkhoff Normal Forms

It is well known that a real analytic symplectic diffeomorphism of the 2d-dimensional disk (dgeq 1) admitting the origin as a non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when d=1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.

On symplectomorphisms and Hamiltonian flows

We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fields, converging to a fixed near the identity C^1 symplectic diffeomorphism psi . This convergence is proved to be uniformly exponentially fast, in a non analytic symplectic topology framework.

Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2

AbstractThe two main results in this paper concern the regularity of the invariant foliation of a $C^0$ -integrable symplectic twist diffeomorphism of the two-dimensional annulus, namely that (i) the generating function of such a foliation is $C^1$ , and (ii) the foliation is Hölder with exponent $\tfrac 12$ . 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 to a rotation. We deduce that every Lipschitz integrable symplectic twist diffeomorphisms of the two-dimensional annulus has Arnol’d–Liouville coordinates and then provide examples of ‘strange’ Lipschitz foliations by smooth curves that cannot be straightened by a symplectic homeomorphism and cannot be invariant by a symplectic twist diffeomorphism.

On the adjoint action of the group of symplectic diffeomorphisms

We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold ($X,\omega$) on $C^\infty(X)$ and on functions $C^\infty(X)\to \mathbb R$. We describe various properties of invariant convex functions on $C^\infty(X)$. Among other things we show that continuous convex functions $C^\infty(X)\to \mathbb R$ that are invariant under the action are automatically invariant under so called strict rearrangements and they are continuous in the sup norm topology of $C^\infty(X)$; but this is not generally true if the convexity condition is dropped.

On the accumulation of separatrices by invariant circles

AbstractLet f be a smooth symplectic diffeomorphism of ${\mathbb R}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if f is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.

Contact Dynamics: Legendrian and Lagrangian Submanifolds

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

Volume-Preserving Diffeomorphisms with the $$\mathcal {M}_0$$-Shadowing Properties

This paper studies the $$\mathcal {M}_0$$ -shadowing property for two types of volume-preserving diffeomorphisms defined on compact manifolds. For symplectic diffeomorphisms defined on symplectic manifolds, the $$C^1$$ -interior of the set of all symplectic diffeomorphisms with the $$\mathcal {M}_0$$ -shadowing property is described by the set of the Anosov diffeomorphisms. If a volume-preserving diffeomorphism in $$\mathrm{Diff}^1_{\mu }(M)$$ is a $$C^1$$ -stable $$\mathcal {M}_0$$ -shadowing diffeomorphism, then M admits a volume-hyperbolic dominated splitting.

Attracted by an elliptic fixed point

We give examples of symplectic diffeomorphisms of R^6 for which the origin is a non-resonant elliptic fixed point which attracts an orbit.

C0-characterization of symplectic and contact embeddings and Lagrangian rigidity

We present a novel [Formula: see text]-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of [Formula: see text]-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from [Formula: see text]-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a [Formula: see text]-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of [Formula: see text]-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

Prequantum transfer operator for symplectic Anosov diffeomorphism

Prequantum transfer operator for symplectic Anosov diffeomorphism

On interrelations between divergence-free and Hamiltonian dynamics

A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold M and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field X with a global cross-section there exist some 4-dimensional symplectic manifold M̃⊃M and a smooth Hamilton function H:M̃→R such that for some c∈R one gets M={H=c} and the Hamiltonian vector field XH restricted on this level coincides with X. For divergence free vector fields with singular points such an extension is impossible but the existence of local cross-section allows one to reduce the dynamics to the study of symplectic diffeomorphisms in some sub-domains of M. We also consider the case of a divergence free vector field X with a smooth integral having only finite number of critical levels. It is shown that such a noncritical level is always a 2-torus and restriction of X on it possesses a smooth invariant 2-form. The linearization of the flow on such a torus (i.e. the reduction to the constant vector field) is not always possible in contrast to the case of an integrable Hamiltonian system but in the analytic case (M and X are real analytic), due to the Kolmogorov theorem, such a linearization is possible for tori with Diophantine rotation numbers.

On symplectic dynamics

This paper contributes to the foundation of various Hofer-like topologies of a closed symplectic manifold (M,ω). Here, considering an arbitrarily linear section of the natural projection of the space of closed 1-forms onto the first de Rham's group, we study various Hofer-like metrics [2,7]. The outcome is that different splitting of the space of closed 1-forms lead to similar “symplectic analogues” of some results from Hamiltonian dynamics: Without appealing to the positivity result of any symplectic displacement energy, we point out an impact of the L∞-Hofer-like lengths in the investigation of the symplectic nature of the uniform limits of sequences of symplectic diffeomorphisms isotopic to the identity map: This provides various symplectic analogues of a result that was proved by Hofer–Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on R2n); and reformulated by Oh–Müller [13] for Hamiltonian diffeomorphisms in general. Moreover, we show that Polterovich's regularization method for Hamiltonian paths extends over the whole group of symplectic paths (no matter the choice of the above section), and use this to prove the equality between the two versions of the corresponding Hofer-like metrics defined on the group of time-one maps of all symplectic isotopies: The symplectic analogues of the uniqueness result of Hofer's geometry [14]. This includes some condition(s) that make the latter uniqueness result continues holds in the context of Bus–Leclercq [7], and a short proof of the non-degeneracy of various Hofer-like metrics. Finally, an alternate proof (without appealing to Floer's theory) of a result from flux geometry found by McDuff–Salamon [12] is given, and various symplectic analogues of some approximation results found by Oh–Müller [13] are provided.

Attracted by an elliptic fixed point

We give examples of symplectic diffeomorphisms of R^6 for which the origin is a non-resonant elliptic fixed point which attracts an orbit.

Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are Cr open and there exists a Cr open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.

On the geometry of the Schmidt-Legendre transformation

Tulczyjew's triples are constructed for the Schmidt-Legendre transformations of both second and third-order Lagrangians. Symplectic diffeomorphisms relating the Ostrogradsky-Legendre and the Schmidt-Legendre transformations are derived. Several examples are presented.

A link between topological entropy and Lyapunov exponents

We show that a $C^{1}$-generic non-partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restricted to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^{1}$-generic set of symplectic diffeomorphisms far from partial hyperbolicity.