33 problems

Let $\Phi$ be a transitive Anosov flow on a $3$-manifold $M$, let $\widetilde{\Phi}$ be its lift to $\widetilde{M}=\mathbb{R}^3$, and let $\mathcal{P}$ be the bifoliated plane asso…

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…

Suppose $n_x>0$, $n_y<0$, and $n_z>0$, and let $M_{n_x,n_y,n_z}$ carry the Anosov flows $\phi_{xy}$ and $\phi_{yz}$ from the two surgery descriptions. A contact-structure swapping…

Let $T^3=\boldsymbol{R}^3/\boldsymbol{Z}^3$ and suppose $n_x>0$, $n_y<0$, and $n_z>0$. Let $M_{n_x,n_y,n_z}$ be the result of the specified surgeries on the coordinate curves, with…

Fried–Christy–Ghys conjecture. Any two transitive Anosov flows with orientable stable and unstable foliations are almost equivalent.

Connectedness conjecture. The space of Anosov metrics on $\Sigma$ is connected.

Uniform spectral gap conjecture. For any $\varepsilon>0$, with probability tending to $1$ as $n\to +\infty$, the cover $M_n\to M$ has no new resonance in the half-space

Let $\varphi_t$ be a smooth, transitive and weakly mixing Anosov flow on a compact manifold $N$. Let $\Gamma$ be a finitely generated torsion-free nilpotent group, let…

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…

Completeness. With such a topology, $\mathcal{TA}\mathcal{F}(M)$ forms a complete metric space, or is complete in a relevant sense.

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…

Positive topological entropy. Every topological Anosov flow $\theta\in\mathcal{TA}\mathcal{F}(M)$ has

Density of periodic orbits. Every topological Anosov flow $\theta\in\mathcal{TA}\mathcal{F}(M)$ possesses a dense set of periodic orbits.

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 $(V,\d7\omega,\d7\lambda)$ be a $4$-dimensional finite type Liouville manifold. Its skeleton $\d7\mathfrak{skel}$ is the set used in the source, and the Liouville flow is the flo…

Let $M$ be a manifold, and consider two oriented Anosov flows on $M$. They are projectively deformation equivalent when they are related through the notion of homotopy through proj…

Let $X$ be an Anosov flow supported by a bicontact structure $(\xi^+,\xi^-)$. For a flow $R$, let $\mathcal{P}(R)$ denote its set of free homotopy data, and let $\mathbb{R}$-covere…

Let $(\xi^+,\xi^-)$ be a bicontact structure supporting an Anosov flow $X$. A Reeb flow is bitransverse when it has the bitransversality property required in the paper, and a flow…

Let $(xi^+,xi^-)$ be a bicontact structure supporting an Anosov flow $X$. The bicontact trichotomy conjecture. Exactly one of the following holds: 1. $xi^+$ is Anosov contact, it a…

Let $M$ be a closed $3$-manifold, let $\varphi$ be a smooth transitive Anosov flow on $M$, and let $\rho$ be an acyclic unitary representation of $\pi_1(M)$. Define the twisted Rue…

Let $M$ be a complete Riemannian manifold with finite volume, whose geodesic flow is Anosov. The periodic-exponent rigidity conjecture. If the unstable Lyapunov exponents are const…

Let $\phi^t$ be a transitive Anosov flow with orientable stable and unstable foliations. A genus one Birkhoff section is a Birkhoff section whose underlying surface has genus one.…

Let $\phi_1^t$ and $\phi_2^t$ be transitive Anosov flows whose stable and unstable line bundles are orientable. Fried–Ghys conjecture. The flows $\phi_1^t$ and $\phi_2^t$ are almos…

Verjovsky's conjecture. Any codimension-one Anosov flow on a closed manifold of dimension greater than three is topologically equivalent to the suspension flow of a hyperbolic tora…

Let $X$ be a transitive Anosov vector field on a closed 3-manifold $\mathcal{M}$. Let $\Omega_0^1$ denote the space of closed resonant $1$-forms at zero, and let $\mathcal{L}_{fX}$…