335 problems

Let $(X_n)_{n\ge 1}$ be a sequence of random variables and let $(a_n)_{n\ge 1}$ be nonnegative numbers, with $A_n:=\sum_{k=1}^n a_k\to\infty$. Assume that there is a constant…

For every function $f$ in the broad class of non-polynomial functions considered by Bergelson, Moreira, and Richter, every $ell\textbackslash in cmathbb{N}$, every invertible measu…

For every $N\ge 2$ and every finitely supported probability measure $\mu$ on $\mathrm{SL}_N(\mathbb{R})$ such that $\Gamma=\langle\operatorname{supp}\mu\rangle$ is discrete and Zar…

For every $1<p\leq\infty$, does there exist a constant $C_p<\infty$, independent of the dimension $d$, such that for every $f\in\ell^p(\mathbb{Z}^d)$,…

For every $1<p\le\infty$, does there exist a constant $C_p<\infty$, independent of the dimension $d$, such that for every $f\in\ell^p(\mathbb{Z}^d)$,…

For every free minimal $\mathbb{Z}^d$-odometer $T$ on a Cantor space and every minimal bounded speedup $S$ of $T$ (in particular, every minimal constant speedup), the actions $S$ a…

Let $G$ be a connected real semisimple Lie group of real rank at least $2$, let $P<G$ be a minimal parabolic subgroup, and let $\Gamma<G$ be a discrete subgroup. If the action…

Let $n\ge 2$, let $(X,\mathcal{B},\mu)$ be a probability space, and let $T_1,\ldots,T_n$ be commuting measure-preserving transformations of $X$. For $f_1,\ldots,f_n\in L^\infty(X)$…

For every dynamical system $(X,T)$ on a compact metric space with zero topological entropy, every $f\in C(X)$, and every $x\in X$, one has…

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…

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…

Positive Rates Conjecture. Every IPS on a one-dimensional lattice with homogeneous interactions of bounded range and positive rates is ergodic.

Let $(X,T)$ be a minimal topological dynamical system, meaning that for every $x\in X$, the set $\{T^n x\}$ is dense, with zero entropy. Let $x_0\in X$, let $f\in C(X)$, and let…

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…

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…

Let $r\geqslant 2$, and let $\mathrm{ITM}(r)$ denote the space of interval translation maps on $r$ intervals. An interval translation map is of infinite type when, for its attracto…

Let $f$ be a conservative partially hyperbolic diffeomorphism of a 3-manifold. Hertz–Hertz–Ures ergodic conjecture. If $f$ is non-ergodic, then there is a 2-torus tangential to…

Sinai's conjecture. For every parameter $\lambda\ne 0$, the metric entropy of $f_\lambda$ is positive. This is presented as another famous positive-entropy conjecture; the source d…

Koopman's irreducibility conjecture. The representation $\pi^{\alpha,\mu,X}$ is irreducible if and only if

Ismagilov's conjecture. The right regular representation is irreducible if and only if

Mahler-measure-zero conjecture. The Mahler measure of the spectrum of $T$ is zero.

Katok's conjecture. Every homogeneous flow on a compact homogeneous space that fails to be stable projects onto a Liouvillean linear flow on a torus. In this case, the flow is stil…

Bergelson–Leibman conjecture. The limit of these averages exists in $L^2$-norm and almost everywhere. The conjecture concerns pointwise and norm convergence of polynomial multiple…

Furstenberg–Bergelson–Leibman conjecture. These averages converge for $\mu$-almost every $x\in X$ as $\min\\{N_1,\ldots,N_k\\}\to\infty$. When $N_1=\cdots=N_k=N$, write the average…

Gatzouras–Peres conjecture. There exists a unique ergodic $T$-invariant measure with the same Hausdorff dimension as $K$. Moreover, this measure is mixing for $T$ and is possibly m…