11 problems

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…

Let $F$ be a smooth vector field on $S^3$ generating a heteroclinic knot $H$, and suppose that $Ess(F)$ with respect to $H$ is finite, possibly empty. An invariant set…

Consider the family of $3$-vector fields $G$ of class $C^2$ exhibiting a singular-hyperbolic attracting set containing some Lorenz-like singularity. Write $s$ for the number of erg…

Let $\mathcal{V}$ be the open family of $C^2$ vector fields admitting a singular-hyperbolic attracting set $\Lambda$. For each $G\in\mathcal{V}$, let $s(G)\in\mathbb{Z}_0^+$ be the…

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…

Let a flow, equivalently a vector field, be given on a manifold. A singular hyperbolic flow is one whose invariant sets admit the corresponding singular-hyperbolic structure; a hom…

Consider the statements of Theorem and the sectional-hyperbolic attracting-set conjecture for $C^2$ flows. $C^1$ regularity conjecture. The statements of Theorem and Conjecture sho…

Higher-order singular-hyperbolicity conjecture. Analogous results should hold for singular hyperbolic sets of every order $p$ with $2\leq p\leq \operatorname{dim}F$. The paper deve…

Let $M$ be a three-dimensional manifold, let ${\cal X}^1(M)$ denote the space of $C^1$ vector fields on $M$, and call a vector field singular hyperbolic when its chain-recurrent se…

Let $M^d$ be a compact manifold, let $X\in {\cal X}^1(M^d)$ be a vector field, and let ${\cal X}^*(M^d)$ denote the set of star vector fields, namely those having a $C^1$ neighborh…

Let $X$ be a vector field. It is star if there is a neighborhood ${\cal U}$ of $X$ such that every critical element of every $Yin{ cal U}$ is hyperbolic; and a vector field is sing…