64 problems

Fried's conjecture. The function $\zeta_{g,\rho}(s)$ is well-defined and nonvanishing at $s=0$, and

Let $f$ be a transitive partially hyperbolic diffeomorphism of a $3$-manifold. A diffeomorphism is a variable time map of a flow if the time spent along each orbit is given by a su…

A flow is codimension one when either its weak-stable or weak-unstable foliation has codimension one. Verjovsky's conjecture. Every smooth codimension-one Anosov flow on a manifold…

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

A flow is a continuous-time dynamical system $(\phi^t)_{t\geq 0}$, and a mixing Anosov flow is an Anosov flow whose correlations decay to zero for suitable observables. Bowen–Ruell…

Let $T^2$ be the $2$-torus, let $f:T^2\to T^2$ be an Anosov map, and consider the suspension flow of $f$. A Goodman–Fried surgery is a surgery operation on a flow of the type refer…

Bonatti–Wilkinson conjecture. Every topological Anosov flow on a closed $3$-manifold is orbitally equivalent to a smooth Anosov flow.

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…

Anosov-flow conjecture. Every robustly transitive action $\varphi\in A^2(\mathbb{R}^2,N)$ with a dense orbit is defined by an Anosov flow, without assuming that the dense orbit is…

Let $\phi_t$ be an Anosov flow with stable foliation $\mathcal{F}_s$. Write $\bar H^p(\mathcal{F}_s)$ for the reduced cohomology of the stable foliation, let…

A projectively Anosov flow is a flow on a closed three-dimensional manifold admitting a projectively Anosov splitting; in the setting considered here, the relevant examples include…

Let a flow be an Axiom A mixing flow, and let correlations be measured with respect to its invariant measure. Exponential-decay conjecture. All Axiom A mixing flows should exhibit…

Let $\phi$ be an Anosov flow on $M$ with orientable stable and unstable foliations and no perfect fits relative to $\mathcal{C}$, and let $\Delta$ be its associated veering triangu…

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.

A transitive Anosov flow is an Anosov flow with a dense orbit. A suspension flow of a hyperbolic linear automorphism of the torus is the suspension flow associated to a hyperbolic…

Resonance-multiplicity conjecture. The following assertions hold:

Generic semisimplicity conjecture. For the trivial representation $\rho=\rho_{\mathrm{triv}}$, the condition

Faure–Weich–Dolgopyat–Pollicott conjecture.

Deterministic high-frequency gap conjecture. For any $\varepsilon>0$, there is a constant $C_\varepsilon$ such that $\mathbf X_{\mathcal E}$ has no resonance in…

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…

Let $\varphi$ be a transitive Anosov flow with transversely orientable stable and unstable foliations. A Dehn–Fried surgery is the surgery operation used in Fried's construction on…

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…