8 problems

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 hyperbolic singularity. Hyperbolic attache…

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…

Genericity conjecture. $\mathcal{T}(X)$ is generic in $\mathcal{C}(X)$ and $\mathcal{S}(X)$.

Bounded stable-set conjecture. There exists $\varepsilon>0$ such that

Let $M$ be a compact surface, and consider the space of $C^1$ area-preserving maps of $M$ with its $C^1$ topology. A map is Anosov if it has a uniformly hyperbolic invariant splitt…

Let $M$ be the compact manifold under consideration, let $f\in\operatorname{Diff}^1(M)$, and let $\mathcal{TLS}$ denote the set of diffeomorphisms with the two-sided limit shadowin…

Plaque expansivity conjecture. Any partially hyperbolic, dynamically coherent diffeomorphism is plaque expansive.

Hammel–Yorke–Grebogi conjecture. A typical dissipative map $f:\mathbb{R}^2\to\mathbb{R}^2$ satisfies