97 problems

Let $X$ be a vector field with flow $\varphi_t$ and nonwandering set $\Omega(X)$. A flow is $\Omega$-stable if its nonwandering dynamics is preserved, up to topological conjugacy,…

A $C^1$-system is a vector field or dynamical system of differentiability class $C^1$. A hyperbolic horseshoe is a compact invariant hyperbolic set on which the dynamics has horses…

Let $M$ be a closed surface and let $\operatorname{Diff}^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$. A diffeomorphism satisfies Axiom A when its nonwandering set is hyp…

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…

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 $M$ be a locally compact Riemannian manifold, let $X\in\mathfrak{X}(M)$ be a smooth vector field, and let $\phi_t$ be its flow. Write $\Omega(\phi_t)$ for the nonwandering set,…

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…

A compact lamination $(L,\mathcal L)$ embedded in a manifold $M$ is preserved by a diffeomorphism $f$ and is normally $AS$ if there exists $\epsilon>0$ such that the saturated nonw…

Let a dynamical system be $C^r$ stable when its limit set is hyperbolic and its stable and unstable manifolds meet transversally at every point. The stability conjecture asserts th…

Let $X$ denote the relevant space of dynamical systems. The structural stability density conjecture asserts that structurally stable dynamical systems are dense among all dynamical…

Consider a dynamical system with quasiperiodic sets, and let a transitive component be a component on which the dynamics is transitive. A trellis is a closed set obtained by closin…

Generic Hyperbolicity Conjecture. Every map $f_0\hskip 2.8in{\cal F}$ can be approximated arbitrarily closely by a hyperbolic map $f\hskip 2.8in{\cal F}$.

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…

Smale's conjecture. Typical dissipative dynamical systems should have dynamics reduced to a finite number of hyperbolic periodic orbits and, in particular, should be structurally s…

Let $(g_k^*,f_k^*)$ be the C-type renormalisation two-cycle, and let $R^2$ denote the second iterate of the renormalisation operator. Consider the relevant parts of the spectrum of…

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…

A star flow is a vector field having a neighborhood in which every vector field has only hyperbolic critical elements. A chain-recurrent class is a dynamically invariant chain-recu…

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…

Let $F$ be a smooth vector field on $S^3$, and let $H$ be a one-dimensional set of the type specified in the paper's classification conjecture. Assume that all fixed points of $F$…

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…