On the accumulation of separatrices by invariant circles

Let 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.

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.

Conley–Zehnder index and bifurcation of fixed points of Hamiltonian maps

We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley–Zehnder index of a fixed point changes. The main result is that when the Conley–Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

Symplectic diffeomorphisms with limit shadowing

Let [Formula: see text] be a symplectic diffeomorphism on a closed [Formula: see text][Formula: see text]-dimensional Riemannian manifold [Formula: see text]. In this paper, we show that [Formula: see text] is Anosov if any of the following statements holds: [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit weak shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the s-limit shadowing property.

Measure expansive symplectic diffeomorphisms and Hamiltonian systems

Let [Formula: see text] be a [Formula: see text]-dimensional ([Formula: see text]), compact smooth Riemannian manifold endowed with a symplectic form [Formula: see text]. In this paper, we show that, if a symplectic diffeomorphism [Formula: see text] is [Formula: see text]-robustly measure expansive, then it is Anosov and a [Formula: see text] generic measure expansive symplectic diffeomorphism [Formula: see text] is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system [Formula: see text] is robustly measure expansive, then [Formula: see text] is Anosov.