15 problems

Let $X\in\mathscr B(\mathbb R^3)$ be a Beltrami field, and let a periodic saddle orbit of $X$ have a homoclinic tangency. A homoclinic tangency is non-degenerately unfolded in…

A vector field $X$ is called multisingular partially hyperbolic if its chain-recurrence set can be split into finitely many compact invariant sets, each admitting a multisingular p…

Newhouse lamination conjecture. Every $d$-dimensional unfolding of a map with a strong homoclinic tangency contains a codimension $2$ Newhouse lamination with Hausdorff dimension…

Let a diffeomorphism be generic and far from homoclinic tangencies, and let chain-recurrence classes be the equivalence classes defined by mutual chain recurrence. Bonatti's conjec…

Let $r\geq 1$ and let $M$ be a three-dimensional manifold. A vector field belongs to ${\cal X}^r(M)$, the space of $C^r$ vector fields on $M$, and a homoclinic tangency is a nontra…

Turaev's conjecture. There is a Newhouse region $C^r$-close to a homoclinic tangency if and only if that tangency is not volume-hyperbolic.

Approximation conjecture. Every three-dimensional flow can be $C^1$ approximated by a flow exhibiting a homoclinic tangency or by a singular-Axiom A flow without cycles.

Let $M$ be a compact manifold, let $\operatorname{Tang}$ denote the set of diffeomorphisms exhibiting a homoclinic tangency, and let…

Let $M$ be a real compact surface, and let $k\geq 1$. For $f\in\mathrm{Diff}^k(M)$, a homoclinic tangency means a tangency between the stable and unstable manifolds of some saddle…

Let $M$ be the compact manifold underlying ${\operatorname{Diff}}^1(M)$, and let $\overline{\rm HT}$ denote the closure of the set of diffeomorphisms with homoclinic tangencies. Bo…

Soit $M$ une variété compacte, et soit $\mathcal{T}$ l'ensemble des difféomorphismes présentant une tangence homocline. Considérons le complémentaire…

Soit $M$ une variété compacte, et notons $\operatorname{Diff}^1(M)$ l'espace des difféomorphismes de classe $C^1$. Une tangence homocline est une intersection non transverse entre…

Soit $M$ une variété compacte, et considérons des difféomorphismes de classe $C^1$ dans $\operatorname{Diff}^1(M)$. Un difféomorphisme est loin des tangences homoclines s'il appart…

Soit $M$ une variété compacte, et notons $\operatorname{Diff}^k(M)$ l'espace des difféomorphismes de classe $C^k$ de $M$. Une tangence homocline est une intersection non transverse…

Let $M$ be a smooth manifold, and let $C^1$-diffeomorphisms of $M$ be considered with the $C^1$ topology. A diffeomorphism has a $C^1$-robust cycle if it exhibits either a homoclin…