10 problems

Palis-type conjecture. The space of closed oriented hypersurfaces in a contact manifold $(Y,\xi)$ decomposes into two open and $C^2$-dense sets: the set of convex hypersurfaces and…

Let a diffeomorphism or non-singular line field be given, with genericity taken in the $C^1$ topology. Palis's conjecture. A $C^1$-generic diffeomorphism (or non-singular line fiel…

Let $M$ be a compact smooth manifold without boundary, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms of $M$. A heterodimensional cycle is a pair of hyper…

Let $M$ be a compact manifold and let $\operatorname{Diff}^1(M)$ be the space of $C^1$-diffeomorphisms. A residual subset is a countable intersection of open dense subsets, and the…

Heterodimensional-cycle extension conjecture. Theorem on possible limit Lyapunov maps for a homoclinic tangency still works if one replaces the homoclinic tangency by a heterodimen…

Let $M$ be a compact manifold and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms. A diffeomorphism is hyperbolic if its nonwandering dynamics has a hyperbolic…

Let $X$ be a vector field. A singular Axiom A vector field without cycle has finitely many chain recurrent classes, each singular hyperbolic; a robustly heterodimensional cycle is…

Let $M$ be a closed manifold, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$-diffeomorphisms of $M$. A robust heterodimensional cycle is a heterodimensional cycle that pe…

Soit $M$ une variété compacte, et notons $\operatorname{Diff}^1(M)$ l'espace des difféomorphismes de classe $C^1$. Un cycle hétérodimensionnel robuste est un cycle hétérodimensionn…

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…