12 problems

Let $\mathcal M$ be a manifold and let $r\geq 1$. Denote by $\operatorname{Diff}^{r}(\mathcal M,\operatorname{vol}_{\mathcal M})$ the class of $C^r$ volume preserving diffeomorphis…

Let $r\ge1$. Consider diffeomorphisms with a dominated splitting, together with non-uniform expansion and non-uniform contraction, and with neither a uniformly expanding nor a unif…

Let $f$ be a diffeomorphism and let the attracting set under consideration have a dominated splitting. Weak non-uniform expansion means that … on a positive-volume subset of points…

Let $\mathcal M$ be a smooth compact manifold. A volume preserving diffeomorphism is nonuniformly hyperbolic if all of its Lyapunov exponents are nonzero at almost every point, and…

Consider a particular parameter value in a smooth dynamical family, and the property that the corresponding map has non-uniform hyperbolicity (or satisfies the weaker conclusion re…

Sufficient-condition conjecture. Then $\Lambda$ is regular.

Eckmann–Ruelle's conjecture. [The supplied statement is truncated before the asserted conclusion.]

Hénon–Lozi tail-index conjectures. For $\phi=\phi_{\alpha}$ with $p_M\in\Lambda$, and for $\phi=\phi_{\theta}$, respectively,

Let $f$ be a nonuniformly hyperbolic map, and let a $d$-pseudotrajectory be a sequence whose one-step errors are less than $d$. For parameters $\theta$ and $\omega$, consider pseud…

The pinball billiard map is considered on its phase space, with dissipation parameter $\lambda$ and Lyapunov exponents $\nu_+$ and $\nu_-$ associated to typical trajectories. Non-u…

Tahzibi's ergodicity conjecture. Every transitive non-uniformly Anosov diffeomorphism is ergodic.

Irregularity conjecture. Generic points of $C^1$-generic diffeomorphisms are irregular$^+$ outside the basins of sinks.