8 problems

Recurrence criterion. The random dynamical system has a unique invariant Borel probability measure if and only if its associated fundamental Markov system is recurrent.

Shub's conjecture. There is a constant $c_{d,k}$ such that

Let $\mathbb{X}^{\circ}$ denote the interior of the state cube, let $\mathbb{F}$ be the collection of faces, and let $F\in\mathbb{F}$ be an attracting face supporting the invariant…

Let $M$ be a closed manifold with a volume, let $k\ge 1$, and let $\operatorname{Diff}^k_{\operatorname{vol}}(M)$ denote the space of $C^k$ volume-preserving diffeomorphisms of $M$…

Let $J$ be the Jacobian of a multiscale, coupled random dynamical system, with state components $\boldsymbol{\eta}$ and $\boldsymbol{\mu}$, and let $H$ denote its Hessian form. A w…

Let $I$ be the phase space and let $T:I\to I$ be a $C^1$, nonsingular dynamical system admitting a unique absolutely continuous invariant measure with positive Lyapunov exponent. L…

Extremal-point conjecture. The set $T$ is a singleton $\{z_0\}$, and the support of the invariant measure that is not supported by one of the two axes is

Cantor attractor conjecture. If $A$ is an interval, the topological attractor $X(w)$ is almost surely a Cantor set.