326 problems

A set $R\subseteq\mathbb{N}_{>0}$ is called a set of nice recurrence if, for every measure-preserving system $(X,\mu,T)$, every measurable set $A\subseteq X$, and every…

For every $n\in\mathbb{N}\cup\{\infty\}$ and every ergodic treeable probability-measure-preserving equivalence relation $\mathcal{R}$ of cost $n$ on a standard probability space…

Let $z=(z_1,\ldots,z_N)$ be a configuration of $N$ equal vortices, each of vorticity $1$, and define the pre-shape set … For almost every $z\in\Gamma_N$, the time average of the em…

Let $\psi_\mathbb{R}$ be a locally Hamiltonian flow on a compact surface $M$ for which Theorem holds. Let $M'$ be one of its minimal components of genus $g\geq 1$, and set … For…

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a strictly increasing sequence. For a system $(X,{\mathcal X},\mu,T)$, say that $a$ is good for seminorm control along $\ell$-term arith…

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence. A sequence is good for $\ell$-step irrational equidistribution if it satisfies the corresponding irrational equidistribution…

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence and let $\ell\in {\mathbb N}$. A sequence is good for mean convergence along $\ell$-term arithmetic progressions if the corre…

Let $G={\rm SL}(n,\mathbb{R})$, $\Gamma={\rm SL}(n,\mathbb{Z})$, and $X=G/\Gamma$ for $n\geq 3$. Let $A<G$ be the group of positive diagonal matrices with determinant one. A Borel…

Veech's conjecture. Condition $\pi_0\perp L^2(\Pi(\kappa))$ for each Furstenberg system $\kappa\in V_S(\boldsymbol{\mu})$ is equivalent to Sarnak's conjecture.

Let $X$ be a separable Banach space that is not isomorphic to a Hilbert space. Recall that $X$ is ergodic if the equivalence relation $\mathbf{E}_0$ is Borel-reducible to the isomo…

Bader–Muchnik irreducibility conjecture. For every locally compact group $G$ and every spread-out probability measure … -boundary of $G$ is irreducible.

Measure-zero conjecture. The subset $M_{\pi,\mathscr R}$ has zero Lebesgue measure in $X_{\pi,\mathscr R}$.

Let $T\in\mathbb{N}$, let $\pi_t$ denote the time-$t$ marginal of the target distribution, and let $G_t$ be the corresponding potential function. Write $P_N$ for the iterated condi…

Let $(X,\mu,T)$ be an ergodic system satisfying the spectral hypothesis of Corollary, and let $(Y,\nu,S)$ be an ergodic nilsystem. A joining of these systems is a joining of the me…

Let $\sigma$ be a primitive substitution of constant length $d$ with eigenvalue $1$ and eigenvector $\gamma$. Write $S_{\lfloor d^n t\rfloor}(\gamma)$ for the associated ergodic su…

Zero-Banach-density sequences conjecture. There exists some $f\in L^1(\mu)$ such that these averages do not converge almost everywhere.

Let $n\geq 2$, let $\phi_1,\ldots,\phi_n$ be a collection of elements of $SU(2)$, and let $L^2_0(SU(2))$ denote the orthogonal complement of the constant functions in $L^2(SU(2))$.…

Euler-component ergodicity conjecture. For each integer $1\leq k\leq 2g+b-2$, the $\mathsf{Mod}_{\Sigma}$-action on the component $e^{-1}(2-2g+b+k)$ of $\mathsf{Hom}(\pi,G)$ is erg…

Free-group ergodicity conjecture. If $r\geq 3$, the action of $\mathsf{Out}(\pi)$ on $\mathsf{Hom}(\pi,G)$ is ergodic.

Campbell–Petersen's conjecture. For every measure-preserving system $(X,\mu,T)$ and every $f\in L^2(X)$,

Let $\Gamma$ be a finitely generated Kleinian group, with limit set $\Lambda$ in the Riemann sphere. Ergodicity conjecture. If … then $\Gamma$ acts ergodically on … This is describ…

Let $\mu_{s,t}$ denote the measure on domino tilings of the plane associated with tilt $(s,t)$. A measure is conditionally uniform when, given a tiling outside any finite region, t…

Joint ergodicity conjecture. The polynomials $p_1,\ldots,p_\ell$ are jointly ergodic for $(X,\mathcal{X},\mu,U)$ if and only if: (1) the product action…

Bergelson–McCutcheon IP-limit positivity conjecture. For every $f\in\mathcal H$,

VIP-system recurrence conjecture. The VIP-system $v$ is good for nice recurrence.