15 problems

Let $\alpha$ be the parameter of the Langford vector field, and let $\alpha_1\approx0.697144898322973$, $\alpha_2\approx0.823$, $\alpha_3\approx0.925$,…

Let $C^r([0,1])$ denote the space of $C^r$ interval maps. For a generic $k$-parameter family $(f_t)_{t\in[-1,1]^k} \subset C^r([0,1])$, where $t\in[-1,1]^k$ is equipped with Lebesg…

Pugh–Shub conjecture. For every $d\geq 1$, a $d$-Kolmogorov typical diffeomorphism displays finitely many sinks and other attractors.

Let $({\mathbb X},d)$ be a metric space, and call $[x,y]:=\{z\in{\mathbb X}:d(x,z)+d(z,y)=d(x,y)\}$ the segment with ends $x,y$. A set is metrically convex if it contains $[x,y]$ f…

Let $F_t$ be a one-parameter iterated function system satisfying properties (H1), (H2), and (H3) of Section 4. If $A_{\bullet}=A^{\bullet}$ for some upper transition attractor of…

Let a sectional-hyperbolic attracting set be $C^1$ close to a sectional-hyperbolic attractor, with no dimensional restrictions; equivalently, allow any combination of stable and ce…

Consider Bianchi type VIII and IX models in the subcritical and critical cases $v\in(0,1/2]$, and let ${\cal A}_-$ denote the attractor as $\tau_-\to\infty$. Define the invariant s…

Consider Bianchi type VIII and IX models in the supercritical case $v\in(1/2,1)$, with stable set $S\subset\mathrm{K}^{\ocircle}$, attractor ${\cal A}_-$, and $\tau_-\to\infty$. Su…

Fractal exponential attractor conjecture. The evolution $(T_t,\mathbf H)$ has a fractal exponential attractor, and

Let $\mathcal{V}$ be the parameter space of finite-type double rotations with a fixed partition, and let the associated map have an attractor. The attractor is semi-continuous with…

Let $M$ be a compact manifold. A property is Arnold–Kolmogorov typical if it holds for Lebesgue almost every small parameter in a Baire generic set of smooth families of dynamics.…

Let $f=f_B$, and let $k$ be sufficiently large depending on the internal parameters $\Omega$, $L$, $b$, and $p$. There is a proper fractal exponential attractor … with finite fract…

Let $H$ be the phase space, let $H^2$ denote the more regular space, let $F$ be the non-linearity satisfying the reasonable assumptions considered in the paper, and let…

Let $M$ be a compact manifold, let ${\rm Diff}^r(M)$ be the space of $C^r$ diffeomorphisms of $M$, and call a point generic when it belongs to the full-measure generic set under co…

Let $M$ be a compact manifold and let ${\rm Diff}^r(M)$ denote the space of $C^r$ diffeomorphisms of $M$. An ergodic attractor is an attractor supporting an ergodic invariant proba…