59 problems

Pugh–Shub conjecture. Stable ergodicity is $mathcal{C}^r$-dense among partially hyperbolic diffeomorphisms.

Sinai's conjecture. For sufficiently large $k$, the standard map $T$ has positive metric entropy with respect to Lebesgue measure.

Buzzi's conjecture. The diffeomorphism $f$ has at most countably many ergodic measures of maximal entropy.

A $C^r$-diffeomorphism is a diffeomorphism of class $C^r$, where $r$ is a real number satisfying $r\geq 2$. A symbolic extension of a dynamical system is a symbolic dynamical syste…

An Anosov diffeomorphism is a uniformly hyperbolic diffeomorphism of a compact manifold. The non-wandering conjecture. Every Anosov diffeomorphism on a compact manifold is non-wand…

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…

Existence conjecture. For each smooth manifold $M$ of dimension $n\ge 2$, there exists a diffeomorphism

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…

An Anosov diffeomorphism is a diffeomorphism of a compact manifold with a uniformly hyperbolic tangent-bundle splitting. An infranilmanifold is a compact manifold finitely covered…

Generic nonlinear reconstruction conjecture. The set $\mathfrak{O}$ of $\delta\in\mathfrak{D}$ such that

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…

Let $S$ be a closed surface, and let $k$-tuples of volume-preserving diffeomorphisms be equipped with the relevant genericity notion. Dolgopyat–Krikorian conjecture. Stable ergodic…

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

Biebler–Berger conjecture. For every $1\le r\le \infty$, there is a nonempty open set $\mathcal B\subset \mathrm{Diff}_{loc}^r(M^2)$ in which the coexistence of infinitely many sin…

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

Let $f$ be a pseudorotation of the closed two-dimensional disk, meaning that it has exactly one periodic point, which is a fixed point. The boundary rotation number is the rotation…

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”…

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…

Liu's conjecture. The measure $\mu$ is an inverse SRB measure if and only if the entropy formula of folding type holds. The paper presents this as an analogue of the characterizati…

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.…

Let $f$ be a $C^2$ volume-preserving diffeomorphism with an $f$-invariant dominated decomposition $TM=E\oplus F$ such that all Lyapunov exponents along $F$ are positive and all Lya…

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 $n\geq 3$, let $\Gamma<\operatorname{SL}(n,{\bf R})$ be a lattice, and let $M$ be a compact manifold of dimension $n$. An action built from tori, $G$-tubes, $G$-disks, blow-ups…