Related papers: Boundaries of instability zones for symplectic twi…

We construct a $C^1$ symplectic twist map f of the annulus that has an essential invariant curve C such that: --C is not differentiable; -- the dynamic of f restricted to C is conjugated to the one of a Denjoy counter-example.

We construct a $C^1$ symplectic twist map $g$ of the annulus that has an essential invariant curve $\Gamma$ such that $\Gamma$ is not differentiable and $g$ restricted to $\Gamma$ is minimal.

For a twist map $f$ of the annulus preserving the Lebesgue measure, we give sufficient conditions to assure the existence of a set of positive measure of points with non-zero asymptotic torsion. In particular, we deduce that every bounded…

The twist interval of a twist map on the annulus $A=\mathbb{T}\times [0,1]$ has nonempty interior if $f$ preserves the area, but could be degenerate for general twist maps. In this note, we show that if a twist map $f$ is non-wandering,…

Let $f$ be a $C^{1+\varepsilon}$ diffeomorphism of the closed annulus $A$ that preserves orientation and the boundary components, and $\widetilde{f}$ be a lift of $f$ to its universal covering space. Assume that $A$ is a Birkhoff region of…

We investigate the regularity of invariant curves of rotation number $1/2$ for a special class of symplectic twist maps of the annulus, billiard maps. We construct strictly convex smooth tables close to the circle having singular (i.e. not…

In four-dimensional symplectic maps complex instability of periodic orbits is possible, which cannot occur in the two-dimensional case. We investigate the transition from stable to complex unstable dynamics of a fixed point under parameter…

We consider a sequences of symplectic twist maps corresponding to Frenkel Kontorova variational functional. For such a sequence without conjugate points we construct an invariant Lagrangian subbundle. Using the ideas by E.Hopf from…

We prove that symplectic twist maps defined on the cotangent bundle of the d-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. the cotangent bundle is continuously foliated by a family of invariant Lagrangian…

We consider a map $F$ of class $C^r$ with a fixed point of parabolic type whose differential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the…

In this paper we consider an analog of the regions of instability for twist maps in the context of area preserving diffeomorphisms which are not twist maps. Several properties analogous to those of classical regions of instability are…

We study the dynamics near infinity of polynomial mappings $f$ in $\mathbb{C}^2$. We assume that $f$ has indeterminacy points and is non constant on the line at infinity $L_\infty$. If $L_\infty$ is $f$-attracting, we decompose the Green…

We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of $\mathbb{T}^2$ in the homotopy class of any linear expanding map with integer eigenvalues. These examples exhibit branching of centre curves along…

In this paper, we consider chaotic dynamics and variational structures of area-preserving maps. There is a lot of study on the dynamics of their maps and the works of Poincare and Birkhoff are well-known. To consider variational structures…

We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the "separation property": different continuity…

For two-parameter families of dissipative twist maps, we investigate the dynamics of invariant graphs as well as the thresholds for their existence and breakdown. Our main results are as follows: (1) For arbitrarily small $C^r$…

We study a class of chainable continua which contains, among others, all inverse limit spaces generated by a single interval bonding map which is piecewise monotone and locally eventually onto. Such spaces are realized as attractors of…

We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.

We consider the homotopy type of maps between symplectic surface whose graphs form symplectic submanifolds of the product. We give a purely topological model for this space in terms of maps with constrained numbers of pre-images. We use…

The key result of this article is key lemma: if a Jordan curve $\gamma$ is invariant by a given C 1+$\alpha$ -diffeomorphism f of a surface and if $\gamma$ carries an ergodic hyperbolic probability $\mu$, then $\mu$ is supported on a…