732 problems

A topological dynamical system is a pair $(X,\tau)$, where $X$ is a compact Hausdorff space and $\tau:X\to X$ is continuous. A bounded sequence $(a_n)_{n\in\mathbb N}$ is determini…

Furstenberg's ×2, ×3 conjecture. Lebesgue measure is the only $T_2,T_3$-invariant nonatomic measure on $\mathbb{R}/\mathbb{Z}$.

Katok's conjecture. The collection of ergodic entropies contains

Let $S$ be a closed oriented surface, and let $\operatorname{Mod}(S)$ be the pure extended mapping class group acting on the character variety $\mathfrak{X}(S)$. The action preserv…

Keane's conjecture. Minimality implies unique ergodicity for interval exchange transformations.

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 $f$ be a volume-preserving partially hyperbolic diffeomorphism of a $3$-manifold with invariant splitting $TM=E^s\oplus E^c\oplus E^u$. A torus is tangent to the…

Let $\Gamma$ be a countable discrete group and let $f\in\mathbf{Z}[\Gamma]$. Let $\boldsymbol{X}_f$ be the algebraic action obtained from the Pontryagin dual of…

Let $V$ be a quadratic stochastic operator on the simplex $S^{m-1}$. It is called ergodic if, for every $\boldsymbol{x}\in S^{m-1}$, the limit … exists. Ulam's ergodic theorem conj…

Let $d,k,n\in\mathbb{N}$, let $T_1,\dots,T_d:X\to X$ be invertible measure-preserving transformations on a probability space $(X,\mathcal{B}(X),\mu)$ that generate a nilpotent grou…

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…

Positive rates conjecture. Every one-dimensional PCA with strictly positive transition rates is ergodic.

Ergodicity conjecture for circular polygons. For certain values of the parameters, the billiard is ergodic.

Let $\alpha$ be a $\mathbb{Z}$-action generated by an ergodic automorphism of a compact abelian group, and let $\zeta_\alpha$ denote its dynamical zeta function. Natural-boundary c…

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…

Let $(X,\mathcal{X},\mu,T_1,\ldots,T_\ell)$ be a system, meaning a standard probability space with commuting invertible measure-preserving transformations, and let…

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…

Let $\mathcal{M}$ be the class of multiplicative functions, and let $f\in\mathcal{M}$ be real-valued. A Furstenberg system of $f$ is a measure-preserving dynamical system arising a…

Eskin–Kontsevich–Zorich regularity conjecture. All ergodic $SL(2,\mathbb{R})$-invariant finite measures $\nu$ on $\mathcal{H}$ are regular. This regularity condition is known for t…

Pierce's conjecture. The discrete analogue of the quadratic Carleson operator is bounded on $\ell^2(\mathbb{Z})$, meaning that there is a constant $C$ such that

Closed-orbit density conjecture. The density of closed orbit measures holds for any transitive piecewise monotonic map with positive topological entropy.

A measure-preserving dynamical system has zero entropy when its Kolmogorov–Sinai entropy is zero. A system is loosely Kronecker when it is loosely Bernoulli; equivalently, in the z…

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

Invariant crossed-product conjecture. The subalgebra $\mathcal{M}$ is a crossed product of the form

Let a horocycle flow be given on a finite-measure dynamical system, and let a sufficiently smooth time change be a smooth reparametrization of that flow. A unitary Koopman represen…