11 problems

Let $f$ be a non-uniformly expanding map. An orbit has a zero Lyapunov exponent if its Lyapunov exponent is zero, and $f$ has exponential decay of correlations if its correlations…

Let $f:M\to M$ be a non-uniformly expanding map with an absolutely continuous invariant measure $\mu$. The map is totally expanding when every invariant measure has all Lyapunov ex…

Consider a generic one-parameter family of maps. Density conjecture. Maps with exponential decay of correlations are Lebesgue density points of other maps with uniform exponential…

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…

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…

Decay of correlations conjecture. The events satisfy $\nu_{L,m,\epsilon}^\mathrm{Lv}(\mathcal A_{x,r,L})\to1$ as first $\epsilon\to0$, then $m\to0$, and finally $L\to\infty$, and,…

High-temperature exponential clustering conjecture. There exist $\beta^*$, $C_\mathrm{Cov}$, $c_\mathrm{Cov}>0$, and $n\in\mathbb{N}$ such that, for every $\beta<\beta^*$ and every…

Let $f$ be the piecewise hyperbolic map under consideration, let $\rho$ be its SRB measure, and let $\rho_{\mathcal S}$ be the slice measure on the singular set. Write $\mathcal M$…

Let $\mathcal L$ be a lattice defining a periodic Lorentz process with phase space $M$, invariant measure $\mu$, free-flight time $\tau$, and flow $\Phi_t$. Let $f,g:M\to\mathbb R$…

Consider smooth singular flows in dimensions higher than three that are sectionally-hyperbolic, meaning that they belong to the higher-dimensional class described in the source, ge…

Let a Lorenz attractor be generated by a flow of regularity $C^{1+\alpha}$. The Lorenz-attractor conjecture. $C^{1+\alpha}$ Lorenz attractors have robust exponential decay of corre…