33 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.

Let ${\mathcal A}$ be the class of probability measures appearing in Theorem, let $\mu$ be an invariant probability measure, and let the theorem's part b) assert uniqueness under t…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be a cardinal. Let $\mu$ be an invariant measure on…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be the relevant cardinal. Let $\mu_0$ and $\mu_1$ be invariant measures on…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be the relevant cardinal. Let $\mu_0$ and $\mu_1$ be invariant measures on…

Consider an annealed time-ergodic invariant measure for the SPDE discussed in the paper. A random field is weakly spatially stationary when it has the weak spatial stationarity pro…

Structural implications of the $d_{\mathrm{An}}$-Cauchy condition. The requirement that $\{X_i\}$ be $d_{\mathrm{An}}$-Cauchy ensures that the limit flow $\theta$ inherits strong s…

Let $X$ be a random walk in Dirichlet environment on ${\mathbb Z}^2$ with positive parameters $(\alpha_1,\ldots,\alpha_4)$. Define … A local accelerating function $\gamma$ is a pos…

Let $\mathbb{T}$ be the unit circle, let $\operatorname{Prob}(\mathbb{T};2,3)$ denote the set of measures invariant under the maps $\omega\mapsto\omega^2$ and…

Consider the dynamical systems constructed from instances of the unique games conjecture, with alphabet size $k$, and let the invariant measure assign weight to neighborhoods of op…

Let ${\bf M}$ be a closed hyperbolic 3-manifold with a totally geodesic surface. Let $h_0=h_0({\bf M})$ be the genus of a totally geodesic surface of smallest genus in ${\bf M}$, a…

Let ${\bf M}$ be a closed hyperbolic 3-manifold. A geometrical limiting measure is a probability measure on ${\mathcal {G}}_2({\bf {M}} )$ obtained as a weak limit of geometrical l…

Keller's conjecture. For every $nuin\ta{P}(X_\varphi)$, there exists $rhoin\ta{P}(Hxtimes\ta{0,1\ta}^{\mathbb{Z}},Rxtimessigma)$ such that, for every measurable $Asubseteq X_\varphi$…

Gaussian absolute-continuity conjecture. If $4\mu$ is a suitable Gaussian measure on $h^{s+\frac{1}{2}}$, then the corresponding measure $4\rho_s$ is absolutely continuous with res…

Natural-extension conjecture. The restriction $\overline{\mathcal{T}_T}:=\mathcal{T}_T|_X$ is bijective Lebesgue almost everywhere, and…

Vanishing-of-cascades conjecture. For every $\alpha s\in{\mathscr S}$, the sequence $\left(\kappa_{\alpha s}(n)\right)_n$ tends to $0$ as $n$ tends to infinity. Consequently, $Q$ i…

Right-eigenvalue conjecture. If, for every $\alpha s\in{\mathscr S}$, the sequence $\left(\kappa_{\alpha s}(n)\right)_n$ converges to $0$, then $Q$ admits $1$ as a right eigenvalue…

Let $\mathfrak{X}$ be the class of maps under consideration, and for each $f\in\mathfrak{X}$ let the acip denote its absolutely continuous invariant probability measure, with densi…

Let $\mathbb{U}_X = \{\,\exp(2i\pi x)\,;x\in X\,\}$ and let $\mathcal{B}$ denote the set of Brjuno numbers. The function … has a continuous extension to $\overline{\mathbb{D}}$ for…

Invariant-measure periodic-orbit conjecture. If $\mu(S^*H)>0$, then $\mu|_{S^*H}$ is supported on a union of periodic geodesics $\gamma$ such that $\gamma\cap S^*H$ has positive ar…

BV stability conjecture. The uniform density is stable with respect to the $BV$-norm for all $\varepsilon>0$, while every other acim is unstable with respect to the $BV$-norm.

Bounded-carrier Brownian invariance conjecture. If $S^K$ is the Brownian motion with drift $c>0$ conditioned to stay within $K$ of its past maximum, then

Atomic-limit conjecture. There exists a sequence of atomic ergodic measures converging to a measure with a positive atomic part.

Let $h_{m,n}^{3,1}(x,y)$ denote the coefficients or derivatives used for the invariant density associated with the region $V_{3,1}$, evaluated at the center $(x,y)$. Vanishing odd-…

Let $h$ be the invariant density, viewed as a function of $x+y\mathbbm{i}$ with real coordinates $x,y$. Consider the points $\pm.5\pm.5\mathbbm{i}$, $\pm.5$, and $\pm.5\mathbbm{i}$…