36 problems

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

Let an AM-system be a pair of increasing affine interval homeomorphisms $\{f_-,f_+\}$ of $[0,1]$, chosen with probabilities $p_-$ and $p_+$, and let $\mu$ be its stationary measure…

Consider the fixed points of the truncator-map dynamics for a system with an increasing number of agents. Fixed-point scarcity conjecture. As the number of agents increases, with o…

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

The stationary density has an asymptotic expansion near the boundary $x_-$ whose higher-order terms include contributions of order $4\mathcal{O}(\ln(x-x_-))4$. Algorithm uses this…

Let $F:\mathcal{X}\to\mathcal{X}$ satisfy $F(0)={\textnormal{D}}F(0)=0$, and let $G:\mathcal{X}\to\mathcal{L}(\mathcal{V},\mathcal{X})$ satisfy the regularity and Lipschitz conditi…

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…

Consider a stable fixed point of the boundary map on the normal bundle of a minimal attractor, and suppose that its eigenvalue stability type changes from real to complex, producin…

Mildly dissipative stationary-measure conjecture. The same conclusion holds for arbitrary mildly dissipative expanding on average systems on surfaces.

DeWitt–expanding conjecture. For a generic tuple, the associated random dynamics is exponentially mixing.

Genericity conjecture. For each closed manifold $M$ and regularity class $k\geq 1$, the expanding on average pairs $(f,g)$ are open and dense in

Let $q_k$ be the limiting cluster probabilities defined by … Vanishing cluster-probability conjecture. We conjecture that $q_k=0$ for all $k$. The contrary could happen if the ball…

The paper develops invariant-manifold theorems for random parabolic evolution equations with almost sectorial operators, including local stable and center manifolds and conditions…

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 $M$…

Necessity conjecture. If the associated Markov chain is not ergodic of degree 2, then

Synchronization conjecture. The set $\tilde{\mathrm{D}}_{\lambda<0}$ contains the whole of $\mathrm{D}$; equivalently,

Let $\Omega$ be the space of discrete subsets of $\mathbb{R}^{2d}$ equipped with the law $\mathbb{Q}$ of a unit-intensity Poisson point process. For a compactly supported, $1$-peri…

Mixing and correlation-decay conjecture. The random systems from Theorem 1.1b are mixing. Moreover, if $\Sigma_R=\{1\}$, then the rate of the annealed decay of correlations is

No absolutely continuous stationary measure conjecture. If $\rho=1$, then $F$ admits no absolutely continuous stationary probability measure.

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

Let $\tau$ be the absorption or exit time of the random dynamical system, let $\tau>t$ denote survival up to time $t$, and let $\mathrm D\varphi_t$ be the linearisation of the flow…

Let $A=(A_1,\dotsc,A_N)\in\mathcal M_d(\mathbb R)^N$. If $\lambda_{\mathrm p}^{\sup}(A)=\lambda_{\mathrm d}(A)$, then there exists $M\in\operatorname{co}(A)$ such that…

Let $A=(A_1,\dotsc,A_N)$ be a tuple of real square matrices, and let $M$ belong to the convex hull of $A$. Suppose that $\lambda(M)=\lambda_{\mathrm d}(A)$, where $\lambda(M)$ deno…

Let $T$ be a periodic map with attracting periodic orbit $\mathcal{O}$, and let $h_\mathbbm{n}$ be the sequence of functions defined in the paper on the interval $I$. For a compact…