46 problems

Let $f$ be a hyperbolic rational map, let $Z(s)$ denote its dynamical zeta function, let \\{\mu_j\} be the zeros of $Z$ counted with multiplicity, and let $\delta$ be the dimensi…

Let $f:M\rightarrow M$ be a $\mathrm{C}^{2}$ diffeomorphism with a compact basic set $\Lambda$ which has associated an almost homoclinic sequence. Let $p\in\Lambda$ be a hyperbolic…

Let a diffeomorphism or non-singular line field be given, with genericity taken in the $C^1$ topology. Palis's conjecture. A $C^1$-generic diffeomorphism (or non-singular line fiel…

A singular star flow is a star flow with singularities, and a singular aperiodic class is a chain-recurrent class containing singularities but no periodic orbit. A homoclinic class…

Effective SPR conjecture. For every $1>\alpha_1>\alpha_2>0$, there exist $\beta_0>0$ and $C>0$ such that, for every ergodic measure $\mu$ of $f$ satisfying

Consider smooth diffeomorphisms with hyperbolic physical measures. A stable leaf has uniform size on a full-volume subset if there is a cylinder in the ambient space on which that…

Equivalence with classical definitions. A continuous flow $\theta$ is a topological Anosov flow according to the source's definition if and only if it satisfies such standard crite…

Structural implications of the $d_{\mathrm{An}}$-Cauchy condition. The requirement that $\{X_i\}$ be $d_{\mathrm{An}}$-Cauchy ensures that the limit flow $\theta$ inherits strong s…

Eckmann–Ruelle conjecture. For any hyperbolic measure $\mu$ of a $C^{1+\alpha}$ diffeomorphism $f$, the pointwise dimension exists almost everywhere and is constant.

Let $F$ be a smooth vector field on $\mathbf{R}^3$ generating a chaotic attractor $A$. Two flows are orbitally equivalent when their trajectories correspond up to a reparametrizati…

Core local product structure conjecture. The set $\operatorname{Core}(\Lambda)$ has local product structure near any regular point, and is locally the product of a Jordan arc by a…

Let $f$ be a smooth diffeomorphism with a hyperbolic set $\Lambda$. For each $x\in\Lambda$, let the stable and unstable slices be the intersections of the local stable and unstable…

Let $M$ be a closed manifold and let $X$ be a $C^1$ vector field on $M$. Let $\Lambda\subset M$ be a chain-recurrent set with an attached hyperbolic singularity. Hyperbolic attache…

Let a hyperbolic set have stable and unstable slices, and interpret “fractal dimension” as either Hausdorff dimension or upper box dimension. Hasselblatt–Schmeling conjecture. The…

Berger's conjecture. There exists a $C^r$-neighborhood $\mathcal{U}$ of $\mathcal{F}$ and an infinite-codimensional subset $\mathcal{N}\subset\mathcal{U}$ such that, for every…

Let $f_0$ and $f_1$ be polynomial automorphisms of ${\mathbb{C}^2}$ with non-trivial dynamics. Write $J_i=J(f_i)$ and $J_i^\star=J^\star(f_i)$ for their Julia sets and assume that…

Let $M$ be the compact manifold underlying the space of diffeomorphisms, and let $\operatorname{Diff}^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$. A saddle-type hyperbol…

Local hyperbolization conjecture. There exists a function $\tilde f\in\mathcal{N}$ such that $p$ is a hyperbolic periodic solution of the equation with nonlinearity $\tilde f$.

Hyperbolic-density conjecture. In the space of polynomials of degree $d>2$ with all critical points real, there are no isentropes of entropy

Let $X$ be a $C^1$ vector field with flow $X_t$, and let $\Gamma$ be a $p$-sectional hyperbolic set with invariant splitting $T_\Gamma M=E\oplus F$. A metric is required to provide…

Let $1\le r\le\infty$ or let $\mathcal C=\mathcal H$, and let $\mathcal C$ be the corresponding category of maps. A $\mathcal C$-diffeomorphism is structurally stable if every…

Let $1\le r\le\infty$ or let $\mathcal C=\mathcal H$, and let $\mathcal C$ be the corresponding category of maps. A $\mathcal C$-diffeomorphism is structurally stable if every…

Let $M$ be a compact manifold. A diffeomorphism of $M$ is Baire generic if it belongs to a countable intersection of open dense sets. Smale's conjecture. A Baire generic diffeomorp…

The square characterization conjecture. For every even integer $p\geq 2$,

Let $M$ be a compact smooth manifold without boundary, and let $\operatorname{Diff}^r(M)$ denote the space of $C^r$ diffeomorphisms of $M$, where $r\geq 1$. An invariant measure is…