24 problems

Let $\Phi$ be an Anosov flow. It is topologically equivalent to a volume-preserving flow when it is topologically equivalent to an Anosov flow that preserves a volume. Volume-prese…

Let $\mathcal{D}$ be the set of circle local homeomorphisms $f:\mathbb{S}^1\to\mathbb{S}^1$ that are topologically conjugate to the doubling map. For $f\in\mathcal{D}$, let $p$ be…

Let $M$ be a closed surface and let $f:M\rightarrow M$ be a $\mathcal{C}^{\infty}$ local diffeomorphism. For an $f$-invariant measure $\mu$, let $\operatorname{HC}(\mu)$ denote the…

Localizability conjecture. There are many spaces of dynamical systems for which any generically localizable property is locally Kolmogorov typical.

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

Let $f$ be a $C^\infty$ mapping, and let $P_f(n)$ denote the number of isolated points of period $n$. In a typical sufficiently parameterized family $(f_a)$, interpret “almost all”…

Let $f$ be a $C^2$-diffeomorphism on a compact Riemannian manifold of dimension greater than one. The map $f$ has the intermediate entropy property when, for every number…

Characterization of smoothness. The flow $\theta$ is topologically conjugate to a smooth Anosov flow generated by some $Y\in\operatorname{An}(M)$ if and only if $X$ belongs to the…

Let $\varphi_{\alpha,b}$ be one of the maps in the setting of Theorem; an SRB/physical measure is a measure with the corresponding SRB or physical property. Uniqueness conjecture.…

Bochi–Katok–Rodriguez Hertz flexibility conjecture. There exists an ergodic diffeomorphism $f \in C$ such that $\xi_i$, for $i=1,\ldots,d$, are the Lyapunov exponents of $f$ with r…

Let $n\geq 3$, let $\Gamma<\operatorname{SL}(n,{\bf R})$ be a lattice, and let $M$ be a compact manifold of dimension $n$. A volume-preserving blow-up or two-sided blow-up is one i…

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 $G$ be a higher-rank simple Lie group, let $\Gamma<G$ be a lattice, let $M$ be a compact manifold, and let $\rho:\Gamma\to\operatorname{Diff}(M)$ be an action. Assume that the…

Let $M$ be a closed hyperbolic 3-manifold, let $\mathcal{F}$ be a $C^2$ taut foliation on $M$, and let $S$ be a fully marked surface. $C^2$ failure conjecture. There exists such a…

Let $M$ be a smooth connected compact manifold, and let a smooth flow be a smooth $\mathbb R$ action on $M$. The flow is globally hypoelliptic if its associated leafwise Laplacian…

Let $M^3$ be a three-dimensional manifold, let $m$ denote Lebesgue measure, and let $\operatorname{Diff}^{1}_{m}(M^3)$ be the space of $C^1$ Lebesgue-measure-preserving diffeomorph…

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 $f$ be a $C^r$-diffeomorphism of a manifold $M$, and let $p$ be a hyperbolic periodic point belonging to a cycle of basic sets having no dominated splitting of any index. Write…

Let $M$ be the connected manifold in the source, let $r>1$, and consider the volume-preserving diffeomorphism space $\operatorname{Diff}^r_m(M)$ with its $C^1$ topology. A diffeomo…

Let $M$ be a compact smooth Riemannian manifold, and let $\operatorname{Diff}^r_m(M)$ denote the $C^r$ diffeomorphisms preserving volume $m$. For $f\in\operatorname{Diff}^r_m(M)$ w…

Let $f\colon M\to M$ be a $C^1$-diffeomorphism, and let $Orb(p)$ be an orbit with well-defined non-vanishing Lyapunov exponents. In the two-dimensional case, assume that … Here…

Let $\Omega$ be a smooth integrable billiard domain, and let $\mathcal E$ be an ellipse with the same perimeter and the same Lazutkin perimeter. Consider the conjugacy between the…

Recall that a strongly simple action has one-dimensional coarse Lyapunov distributions, which integrate to Lyapunov foliations, and that some Lyapunov foliations have conditional m…

Let $r>1$, let $M$ be a compact manifold of dimension $d$, and let $T:M\to M$ be a $^r$ map. For nonintegral $r$, assume $T\in \mathcal{C}^{\lfloor r\rfloor}$ and…