20 problems

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…

Palis's conjecture. The space of two-dimensional $C^r$ diffeomorphisms with $r\ge 2$ is the closure of the union of just two open sets: axiom A systems and the Newhouse domain. Thi…

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…

A homoclinic tangency is a nontransverse intersection between the stable and unstable manifolds of a hyperbolic periodic point. Consider diffeomorphisms of class $C^r$, with…

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 $M$ be a compact boundaryless manifold, let $\operatorname{Diff}^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$, and let $\mathcal{T}$ be the set of diffeomorphisms exh…

Palis's finite-attractors conjecture. With total probability in the parameter line, the corresponding diffeomorphisms do not display infinitely many attractors, in particular sinks…

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 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…

Homoclinic-tangency approximation conjecture. Any diffeomorphism exhibiting infinitely many attractors can be approximated by a diffeomorphism which exhibits a homoclinic tangency.

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…

Bonatti's tameness conjecture. The set ${\cal O}_0(M)$ consists of tame diffeomorphisms. The source notes that ${\cal O}_0(M)$ is nonempty and that it is an open question whether i…

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…

A diffeomorphism is a smooth dynamical system on a manifold. A homoclinic bifurcation is either a homoclinic tangency or a heterodimensional cycle. Palis's denseness conjecture. Ho…