15 problems

Abdenur–Diaz conjecture. A $C^1$-generic diffeomorphism with the shadowing property is structurally stable.

Hammel–Yorke–Grebogi's conjecture. A typical dissipative map $f:\mathds{R}^2\to\mathds{R}^2$ with positive Lyapunov exponent satisfies $\operatorname{FinHolSh}(1/2,1/2)$. This conj…

Let $M$ be a closed differentiable manifold, and let $f\colon M\to M$ be a $C^1$-generic diffeomorphism. Denote by $\Omega(f)$ the non-wandering set of $f$. Arbieto–Bonatti–Diaz co…

Generalized hyperbolicity conjecture. An invertible linear operator exhibits the shadowing property if and only if it is generalized hyperbolic.

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…

Let $F$ be a free group of finitely many generators acting on a manifold $M$. Free-group shadowing conjecture. The action of $F$ on $M$ cannot have the shadowing property. This con…

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

Let $M$ be a closed topological manifold and let $f:M\to M$ be a homeomorphism. Denote by $\Omega(f)$ the non-wandering set of $f$, and let $f|_{\Omega(f)}$ be the restriction of…

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…

Existence conjecture. There exist nonempty open subsets of $C^1$-stably weakly shadowing symplectomorphisms of dimension at least $6$ such that $M$ is not uniformly hyperbolic for…

Let $f$ be a nonuniformly hyperbolic map, and let a $d$-pseudotrajectory be a sequence whose one-step errors are less than $d$. For parameters $\theta$ and $\omega$, consider pseud…

Let a numerical trajectory be generated by a molecular-dynamics discretization over a long time interval, and let an exact trajectory solve the original differential equations. Sha…