7 problems

Consider the relevant higher-dimensional phase space and its dynamical map; an orbit is dense when its closure is the whole phase space. Mather's transitivity conjecture. Generical…

Let $M$ be the compact manifold underlying $Diff^1(M)$, let $Diff^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$, and let $\Omega(f)$ be the non-wandering set of $f$. A sub…

Let $f(X,Y)$ be the performance difference function for random traits $X$ and $Y$, and let $\boldsymbol{\nabla} f$ denote its gradient. Suppose that the gradient is nonzero almost…

Gogolev–Maimon–Kolgomorov conjecture. For all analytic diffeomorphisms $f$ in a sufficiently small neighborhood of $A$, the strong unstable foliation ${\mathcal W}^{uu}$ is transit…

Abdenur–Bonatti–Díaz conjecture. $C^1$-generic transitive diffeomorphisms whose non-wandering set has a non-empty interior are globally transitive.

Let $M$ be a compact connected smooth Riemannian manifold. A $C^1$ diffeomorphism $\phi\colon M\to M$ is Anosov if there are constants $0<\lambda<1$, $K>0$, and a $D\phi$-invariant…

Theorem 1 concerns a compact orientable surface $S$ with a smooth volume form and asserts that a residual subset of pairs in…