15 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…

Generic appearance conjecture. Coindex-1 heterodimensional cycles can appear, with very few exceptions, in any homoclinic or heteroclinic bifurcation whose effective dimension allo…

Hyperbolicity–wildness–heterodimensional dynamics conjecture. Every diffeomorphism is either uniformly hyperbolic, or it is arbitrarily close in $C^r$ to a diffeomorphism with wild…

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…

Let $M$ be a compact manifold and let $\operatorname{Diff}^1(M)$ be the space of $C^1$-diffeomorphisms. A heterodimensional cycle is a pair of hyperbolic periodic points with diffe…

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…

Bonatti's conjecture. Every robustly non-hyperbolic diffeomorphism could be approximated by a diffeomorphism which has a robust heterodimensional cycle.

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…

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…