15 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…

A singular star flow is a star flow with singularities, and a singular aperiodic class is a chain-recurrent class containing singularities but no periodic orbit. A homoclinic class…

A star flow is a vector field having a neighborhood in which every vector field has only hyperbolic critical elements. A chain-recurrent class is a dynamically invariant chain-recu…

Let $F$ be a continuous-time dynamical system with a compact global attractor, and let $f=F^T$ be its time-$T$ map for some $T>0$. A point $y$ is a chain-limit point of $x$ if it c…

Let $M$ be a closed manifold and let $X$ be a $C^1$ vector field on $M$. Let $\Lambda\subset M$ be a chain-recurrent set with an attached singularity. Attached-singularity non-shad…

Let $X$ be a vector field and let its chain recurrent set be the set of points returning to themselves by arbitrarily small pseudo-orbits. A vector field is a star vector field if…

Ma{n}'e's conjecture. If a diffeomorphism satisfies the star condition, then it is hyperbolic, that is, its chain recurrent set is hyperbolic.

Preperiodicity conjecture. For $r\geq 1$, there exists a $C^r$-residual set of diffeomorphisms for which any invariant measure supported in a chain-recurrent set is $C^r$-preperiod…

Generic mechanical non-domination conjecture. There is a residual subset $\mathcal R$ of $C^r$-diffeomorphisms $f$ such that if $\mathcal C(P,f)$ is not dominated of index $i$, the…

Let $M$ be a compact manifold, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms. A chain-recurrence class is a sink when it is represented by an attracting…

Let $M$ be the underlying compact manifold, let ${\cal X}^1(M)$ denote the space of $C^1$ vector fields, and let $X\in{\cal X}^1(M)$. A vector field is approximable by homoclinic t…

Let $X$ be a generic vector field, and let $C(\sigma)$ be a non-trivial chain recurrent class containing a hyperbolic singularity $\sigma$. A chain recurrent class is non-trivial w…

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