31 problems

Standard map entropy conjecture. The map $f_{\lambda}$ has positive metric entropy for every $\lambda\neq 0$. Despite its simple formulation and extensive study, this conjecture re…

Let $M$ be a closed Riemannian manifold without conjugate points, and let the Liouville measure denote the natural invariant probability measure for its geodesic flow. Freire–Mañé'…

Let $K$ and $T$ be symmetric convex bodies in $\mathbb{R}^n$, and let $\gamma_p(K,T)$ denote Talagrand's functional associated with the metric induced by $T$. The preceding theorem…

Let $D$ denote the Euclidean unit ball in $\mathbb{R}^n$, and let $M^*(A)$ denote the mean width functional used in the paper. For a set $S=\{x_i\}_{i=1}^{2^k}\subset\mathbb{R}^n$,…

Let $K$ be a convex body that is the convex hull of at most $2^k$ points in $R D$ and satisfies $N(K,D)\le 2^k$. Let $\gamma(K)$ be the parameter defined for this class of bodies i…

Let $X$ and $Y$ be Banach spaces with unit balls $B(X)$ and $B(Y)$, respectively, and let $u:X\to Y$ be a linear operator. Define its $k$th entropy number by … Let $u^*:Y^*\to X^*$…

Let $K$ and $T$ be symmetric convex bodies in $\mathbb{R}^n$, and let $N(A,B)$ denote the covering number of $A$ by translates of $B$. For a convex body $A$, write $A^{\circ}$ for…

Let $K\subset\mathbf R^n$ be a symmetric convex body, let $\mathrm{GL}(n)$ denote the group of invertible linear maps on $\mathbf R^n$, and let $M$ and $M^\ast$ be the mean width a…

Let a bounded real-valued function class have finite $gamma$-fat-shattering dimension, and consider the covering-number or metric-entropy bound obtained at the relevant scale. **R…

Let $gamma0$ and let $the gamma$-fat-shattering dimension of a bounded real-valued function class be finite. Consider the metric entropy of the class at the relevant scale, measu…

Let $f$ be a surface symplectomorphism, and interpret “typical” in the conjectural generic or parameterwise sense under discussion. Positive metric entropy conjecture. A typical su…

Let $X$ be a function space or linear image whose entropy is bounded using low-dimensional approximations, and suppose the corresponding bounds have logarithmic factors. Log-term c…

Conjecture on superfluous logarithmic terms. There are indeed superfluous log-terms in the bounds obtained in this work.

Let $V$ be a $(K,n)$ regular set in $\mathcal{B}^N$, and let $\mathcal{N}(V,\eps)$ denote its covering number at scale $\eps$. Covering-number dependence conjecture. There exists a…

Let $T\colon[0,1)\to[0,1)$ be the Bolyai–Rényi transformation, and let $\mu$ be its unique $T$-invariant probability measure. The Bosma–Dajani–Kraaikamp conjecture. The metric entr…

Let $S$ be the function class and $\mathcal{X}$ the domain, with $\mathcal{N}(S(\mathcal{X}),4\epsilon,|\cdot|_\infty)$ denoting the covering number of the restricted function clas…

Let $S_1$ and $S_2$ be the Lagrangian spheres under consideration, let $\tau_1$ and $\tau_2$ be the corresponding symplectic Dehn twists, and let $P_\psi(S_1,S_2)$ denote their ass…

Let $S_1$ and $S_2$ be Lagrangian $n$-spheres in an $A_2^n$-configuration, intersecting transversely at a single point, and let $\tau_1$ and $\tau_2$ be symplectic Dehn twists alon…

Let $f:\mathbb T^2\to\mathbb T^2$ be the standard map … Here $\epsilon>0$ and Lebesgue measure is the natural volume measure on $\mathbb T^2$. The standard map positive metric entr…

Let ${\upmu}$ denote the norm used to measure functions in the metric-entropy conditions, and let RKHS classes be the function classes under consideration. Farahmand et al.'s conje…

Let $(X,d)$ be a topological dynamical system with the periodic specification property, and let $f$ be its dynamics. An $f$-invariant measure is a Borel probability measure invaria…

Let $M$ be a compact oriented Riemannian manifold with smooth volume $bc$, and let $PH^{2,\frac 12+}_{\mu,1}(M)\setminus\mathcal A$ denote the specified class of diffeomorphisms fr…

Let $M$ be a compact manifold, let $m$ denote Lebesgue measure, and let $\operatorname{Diff}^{1}_{m}(M)$ be the space of $C^1$ Lebesgue-measure-preserving diffeomorphisms. A diffeo…

Let $\mathbb M$ be a surface with area form $\omega$, and let $C^r_\omega$-generic mean generic in the space of $C^r$ area-preserving surface diffeomorphisms. Generic entropy dicho…

Herman's positive entropy conjecture. For every $\varepsilon>0$ there exists $f\in {\rm Diff}^\infty_\omega(\mathbb D)$ such that $d_\infty(f,id)<\varepsilon$ and